Search: a005266 -id:a005266
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A000058
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Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.
(Formerly M0865 N0331)
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+10
107
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OFFSET
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0,1
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COMMENTS
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Also called Euclid numbers, because a(n) = a(0)*a(1)*...*a(n-1) + 1. - Jonathan Sondow, Jan 26 2014
Another version of this sequence is given by A129871, which starts with 1, 2, 3, 7, 43, 1807, ... .
The greedy Egyptian representation of 1 is 1 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1807 + ... .
Take a square. Divide it into 2 equal rectangles by drawing a horizontal line. Divide the upper rectangle into 2 squares. Now you can divide the lower one into another 2 squares, but instead of doing so draw a horizontal line below the first one so you obtain a (2+1 = 3) X 1 rectangle which can be divided in 3 squares. Now you have a 6 X 1 rectangle at the bottom. Instead of dividing it into 6 squares, draw another horizontal line so you obtain a (6+1 = 7) X 1 rectangle and a 42 X 1 rectangle left, etc. - Néstor Romeral Andrés, Oct 29 2001
More generally one may define f(1) = x_1, f(2) = x_2, ..., f(k) = x_k, f(n) = f(1)*...*f(n-1)+1 for n > k and natural numbers x_i (i = 1, ..., k) which satisfy gcd(x_i, x_j) = 1 for i <> j. By definition of the sequence we have that for each pair of numbers x, y from the sequence gcd(x, y) = 1. An interesting property of a(n) is that for n >= 2, 1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n-1) = (a(n)-2)/(a(n)-1). Thus we can also write a(n) = (1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n-1) - 2 )/( 1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n-1) - 1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), May 10 2001; [corrected by Michel Marcus, Mar 27 2019]
A greedy sequence: a(n+1) is the smallest integer > a(n) such that 1/a(0) + 1/a(1) + 1/a(2) + ... + 1/a(n+1) doesn't exceed 1. The sequence gives infinitely many ways of writing 1 as the sum of Egyptian fractions: Cut the sequence anywhere and decrement the last element. 1 = 1/2 + 1/3 + 1/6 = 1/2 + 1/3 + 1/7 + 1/42 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = ... . - Ulrich Schimke, Nov 17 2002; [corrected by Michel Marcus, Mar 27 2019]
Consider the mapping f(a/b) = (a^3 + b)/(a + b^3). Starting with a = 1, b = 2 and carrying out this mapping repeatedly on each new (reduced) rational number gives 1/2, 1/3, 4/28 = 1/7, 8/344 = 1/43, ..., i.e., 1/2, 1/3, 1/7, 1/43, 1/1807, ... . Sequence contains the denominators. Also the sum of the series converges to 1. - Amarnath Murthy, Mar 22 2003
a(1) = 2, then the smallest number == 1 (mod all previous terms). a(2n+6) == 443 (mod 1000) and a(2n+7) == 807 (mod 1000). - Amarnath Murthy, Sep 24 2003
An infinite coprime sequence defined by recursion.
Apart from the initial 2, a subsequence of A002061. It follows that no term is a square.
It appears that a(k)^2 + 1 divides a(k+1)^2 + 1. - David W. Wilson, May 30 2004. This is true since a(k+1)^2 + 1 = (a(k)^2 - a(k) + 1)^2 +1 = (a(k)^2-2*a(k)+2)*(a(k)^2 + 1) (a(k+1)=a(k)^2-a(k)+1 by definition). - Pab Ter (pabrlos(AT)yahoo.com), May 31 2004
In general, for any m > 0 coprime to a(0), the sequence a(n+1) = a(n)^2 - m*a(n) + m is infinite coprime (Mohanty). This sequence has (m,a(0))=(1,2); (2,3) is A000215; (1,4) is A082732; (3,4) is A000289; (4,5) is A000324.
Any prime factor of a(n) has -3 as its quadratic residue (Granville, exercise 1.2.3c in Pollack).
Note that values need not be prime, the first composites being 1807 = 13 * 139 and 10650056950807 = 547 * 19569939581. - Jonathan Vos Post, Aug 03 2008
If one takes any subset of the sequence comprising the reciprocals of the first n terms, with the condition that the first term is negated, then this subset has the property that the sum of its elements equals the product of its elements. Thus -1/2 = -1/2, -1/2 + 1/3 = -1/2 * 1/3, -1/2 + 1/3 + 1/7 = -1/2 * 1/3 * 1/7, -1/2 + 1/3 + 1/7 + 1/43 = -1/2 * 1/3 * 1/7 * 1/43, and so on. - Nick McClendon, May 14 2009
(a(n) + a(n+1)) divides a(n)*a(n+1)-1 because a(n)*a(n+1) - 1 = a(n)*(a(n)^2 - a(n) + 1) - 1 = a(n)^3 - a(n)^2 + a(n) - 1 = (a(n)^2 + 1)*(a(n) - 1) = (a(n) + a(n)^2 - a(n) + 1)*(a(n) - 1) = (a(n) + a(n+1))*(a(n) - 1). - Mohamed Bouhamida, Aug 29 2009
This sequence is also related to the short side (or hypotenuse) of adjacent right triangles, (3, 4, 5), (5, 12, 13), (13, 84, 85), ... by A053630(n) = 2*a(n) - 1. - Yuksel Yildirim, Jan 01 2013, edited by M. F. Hasler, May 19 2017
For n >= 4, a(n) mod 3000 alternates between 1807 and 2443. - Robert Israel, Jan 18 2015
The set of prime factors of a(n)'s is thin in the set of primes. Indeed, Odoni showed that the number of primes below x dividing some a(n) is O(x/(log x log log log x)). - Tomohiro Yamada, Jun 25 2018
Sylvester numbers when reduced modulo 864 form the 24-term arithmetic progression 7, 43, 79, 115, 151, 187, 223, 259, 295, 331, ..., 763, 799, 835 which repeats itself until infinity. This was first noticed in March 2018 and follows from the work of Sondow and MacMillan (2017) regarding primary pseudoperfect numbers which similarly form an arithmetic progression when reduced modulo 288. Giuga numbers also form a sequence resembling an arithmetic progression when reduced modulo 288. - Mehran Derakhshandeh, Apr 26 2019
Named after the English mathematician James Joseph Sylvester (1814-1897). - Amiram Eldar, Mar 09 2024
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REFERENCES
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Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
Richard K. Guy and Richard Nowakowski, Discovering primes with Euclid. Delta, Vol. 5 (1975), pp. 49-63.
Amarnath Murthy, Smarandache Reciprocal partition of unity sets and sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Mehran Derakhshandeh, Why do Sylvester numbers, when reduced modulo 864, form an arithmetic progression 7,43,79,115,151,187,223,...?
Richard K. Guy and Richard Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, Vol. 124, No. 3 (2017), pp. 232-240, arXiv preprint, arXiv:math/1812.06566 [math.NT], 2018.
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FORMULA
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a(n) = 1 + a(0)*a(1)*...*a(n-1).
Vardi showed that a(n) = floor(c^(2^(n+1)) + 1/2) where c = A076393 = 1.2640847353053011130795995... - Benoit Cloitre, Nov 06 2002 (But see the Aho-Sloane paper!)
a(n) = f(1/(1-(1/a(0) + 1/a(1) + ... + 1/a(n-1)))) where f(x) is the smallest integer > x (see greedy algorithm above). - Robert FERREOL, Feb 22 2019
Sum_{n>=0} (-1)^n/(a(n)-1) = A118227.
Sum_{n>=0} (-1)^n/a(n) = 2 * A118227 - 1. (End)
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EXAMPLE
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a(0)=2, a(1) = 2+1 = 3, a(2) = 2*3 + 1 = 7, a(3) = 2*3*7 + 1 = 43.
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MAPLE
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A[0]:= 2:
for n from 1 to 12 do
A[n]:= A[n-1]^2 - A[n-1]+1
od:
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MATHEMATICA
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a[0] = 2; a[n_] := a[n - 1]^2 - a[n - 1] + 1; Table[ a[ n ], {n, 0, 9} ]
RecurrenceTable[{a[n + 1] == a[n]^2 - a[n] + 1, a[0] == 2}, a, {n, 0, 10}] (* Emanuele Munarini, Mar 30 2017 *)
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PROG
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(PARI) a(n)=if(n<1, 2*(n>=0), 1+a(n-1)*(a(n-1)-1))
(PARI) A000058(n, p=2)={for(k=1, n, p=(p-1)*p+1); p} \\ give Mod(2, m) as 2nd arg to calculate a(n) mod m. - M. F. Hasler, Apr 25 2014
(PARI) a=vector(20); a[1]=3; for(n=2, #a, a[n]=a[n-1]^2-a[n-1]+1); concat(2, a) \\ Altug Alkan, Apr 04 2018
(Haskell)
a000058 0 = 2
a000058 n = a000058 m ^ 2 - a000058 m + 1 where m = n - 1
(Haskell)
(Python)
for n in range(1, 10):
(Maxima) a(n) := if n = 0 then 2 else a(n-1)^2-a(n-1)+1 $
(Julia)
a(n) = n == 0 ? BigInt(2) : a(n - 1)*(a(n - 1) - 1) + 1
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CROSSREFS
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Cf. A005267, A000945, A000946, A005265, A005266, A075442, A007018, A014117, A054377, A002061, A118227, A126263, A007996 (primes dividing some term), A323605 (smallest prime divisors), A129871 (a variant starting with 1).
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KEYWORD
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nonn,nice,core
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AUTHOR
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STATUS
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approved
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A000945
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Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime factor of 1 + Product_{k=1..n} a(k).
(Formerly M0863 N0329)
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+10
101
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2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357
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OFFSET
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1,1
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COMMENTS
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"Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski and later by Shanks, [Wagstaff 1993] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored. Already the number a(1)* ... *a(43) + 1 has 180 digits." - Crandall and Pomerance
If this variant of Euclid-Mullin sequence is initiated either with 3, 7 or 43 instead of 2, then from a(5) onwards it is unchanged. See also A051614. - Labos Elemer, May 03 2004
Wilfrid Keller informed me that a(1)* ... *a(43) + 1 was factored as the product of two primes on Mar 09 2010 by the GNFS method. See the post in the Mersenne Forum for more details. The smaller 68-digit prime is a(44). Terms a(45)-a(47) were easy to find. Finding a(48) will require the factorization of a 256-digit number. See the b-file for the four new terms. - T. D. Noe, Oct 15 2010
On Sep 11 2012, Ryan Propper factored the 256-digit number by finding a 75-digit factor by using ECM. Finding a(52) will require the factorization of a 335-digit number. See the b-file for the terms a(48) to a(51). - V. Raman, Sep 17 2012
A056756 gives the position of the k-th prime in this sequence for each k. - Jianing Song, May 07 2021
Named after the Greek mathematician Euclid (flourished c. 300 B.C.) and the American engineer and mathematician Albert Alkins Mullin (1933-2017). - Amiram Eldar, Jun 11 2021
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REFERENCES
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Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 6.
Richard Guy and Richard Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, Vol. 8 (1993), pp. 23-32.
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LINKS
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Richard Guy and Richard Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
Lucas Hoogendijk, Prime Generators, Bachelor Thesis, Utrecht University (Netherlands, 2020).
Thorkil Naur, Letter to N. J. A. Sloane, Aug 27 1991, together with copies of "Mullin's sequence of primes is not monotonic" (1984) and "New integer factorizations" (1983) [Annotated scanned copies]
Samuel S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, Vol. 8 (1993), pp. 23-32. (Annotated scanned copy)
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EXAMPLE
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a(5) is equal to 13 because 2*3*7*43 + 1 = 1807 = 13 * 139.
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MAPLE
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a :=n-> if n = 1 then 2 else numtheory:-divisors(mul(a(i), i = 1 .. n-1)+1)[2] fi: seq(a(n), n=1..15);
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MATHEMATICA
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f[1]=2; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[1, 1]]; Table[f[n], {n, 1, 46}]
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PROG
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(PARI) P=[]; until(, print(P=concat(P, factor(vecprod(P)+1)[1, 1]))) \\ Jeppe Stig Nielsen, Apr 01 2024
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CROSSREFS
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KEYWORD
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nonn,nice,hard,changed
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AUTHOR
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STATUS
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approved
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A000946
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Euclid-Mullin sequence: a(1) = 2, a(n+1) is the largest prime factor of 1 + Product_{k=1..n} a(k).
(Formerly M0864 N0330)
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+10
53
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2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, 20766142440959799312827873190033784610984957267051218394040721
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OFFSET
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1,1
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COMMENTS
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Booker's abstract claims: "We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten."
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REFERENCES
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R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
Thorkil Naur, Letter to N. J. A. Sloane, Aug 27 1991, together with copies of "Mullin's sequence of primes is not monotonic" (1984) and "New integer factorizations" (1983) [Annotated scanned copies]
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)
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MATHEMATICA
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f[1] = 2; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[-1, 1]]; Table[f[n], {n, 1, 10}] (* Alonso del Arte, Jun 25 2011 based on the program given for A000945 *)
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PROG
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(PARI) gpf(n)=my(f=factor(n)[, 1]); f[#f];
first(m)=my(v=vector(m)); v[1]=2; for(i=2, m, v[i]=gpf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 14 2015
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CROSSREFS
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KEYWORD
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nonn,nice,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A005265
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a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the smallest prime factor of b(n)-1.
(Formerly M2246)
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+10
47
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3, 2, 5, 29, 11, 7, 13, 37, 32222189, 131, 136013303998782209, 31, 197, 19, 157, 17, 8609, 1831129, 35977, 508326079288931, 487, 10253, 1390043, 18122659735201507243, 25319167, 9512386441, 85577, 1031, 3650460767, 107, 41, 811, 15787, 89, 68168743, 4583, 239, 1283, 443, 902404933, 64775657, 2753, 23, 149287, 149749, 7895159, 79, 43, 1409, 184274081, 47, 569, 63843643
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OFFSET
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1,1
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COMMENTS
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Suggested by Euclid's proof that there are infinitely many primes.
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REFERENCES
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R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.
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LINKS
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R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)
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MAPLE
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a :=n-> if n = 1 then 3 else numtheory:-divisors(mul(a(i), i = 1 .. n-1)-1)[2] fi:seq(a(n), n=1..15);
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PROG
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(PARI) lpf(n)=factor(n)[1, 1] \\ better code exists, usually best to code in C and import
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CROSSREFS
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KEYWORD
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nonn,nice,hard
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AUTHOR
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STATUS
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approved
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A051308
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Euclid-Mullin sequence (A000945) with initial value a(1)=5 instead of a(1)=2.
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+10
39
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5, 2, 11, 3, 331, 19, 199, 53, 21888927391, 29833, 101, 71, 23, 311, 7, 72353, 13, 227, 96014559769, 5641, 41, 82107739003, 67, 169637539, 61, 29, 31319, 17, 97, 238591921, 313, 102065429, 157, 37, 595553520313, 244217, 241, 4773229353714971081083834237, 103
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OFFSET
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1,1
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COMMENTS
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The initial primes 3 and 7 give essentially A000945.
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LINKS
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EXAMPLE
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5*2*11*3 + 1 = 331, which is prime; the least prime factor of 330*331 + 1 = 109231 = 19*5749 is 19, so a(6) = 19.
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MATHEMATICA
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a[1]=5; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
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PROG
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(PARI) spf(n)=my(f=factor(n)[1, 1]); f;
first(m)=my(v=vector(m)); v[1]=5; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; \\ Anders Hellström, Aug 15 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A051335
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Euclid-Mullin sequence (A000945) with initial value a(1)=127 instead of a(1)=2.
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+10
33
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127, 2, 3, 7, 5, 149, 19, 41, 23899, 139, 43, 761, 281, 17, 53, 2551, 23, 20149, 100720363856036298033578901613089271, 31, 179, 11, 13, 523, 282995646721, 2871347, 83, 10744429, 1031, 427773048135533, 97, 78506876242349, 67
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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a[1]=127; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
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PROG
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(PARI) spf(n)=my(f=factor(n)[1, 1]); f;
first(m)={my(v=vector(m)); v[1]=127; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; } /* Anders Hellström, Aug 18 2015 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A057204
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Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.
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+10
28
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7, 199, 7761799, 487, 67, 103, 3562539697, 7251847, 13, 127, 5115369871402405003, 31, 697830431171707, 151, 3061, 229, 193, 5393552285540920774057256555028583857599359699, 709, 397, 37, 61, 46168741, 3127279, 181, 122268541
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OFFSET
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1,1
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COMMENTS
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4*Q^2 + 3 always has a prime divisor congruent to 1 modulo 6.
If we start with the empty product Q=1 then it is not necessary to specify the initial prime. - Jens Kruse Andersen, Jun 30 2014
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REFERENCES
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P. G. L. Dirichlet (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.
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LINKS
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EXAMPLE
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a(4)=487 is the smallest prime divisor of 4*Q*Q + 3 = 10812186007, congruent to 1 (mod 6), where Q = 7*199*7761799.
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MATHEMATICA
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a={7}; q=1;
For[n=2, n<=7, n++,
q=q*Last[a];
AppendTo[a, Min[Select[FactorInteger[4*q^2+3][[All, 1]], Mod[#, 6]==1 &]]];
];
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PROG
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(PARI) Q=1; for(n=1, 11, f=factor(4*Q^2+3); for(i=1, #f~, p=f[i, 1]; if(p%6==1, break)); print1(p", "); Q*=p) \\ Jens Kruse Andersen, Jun 30 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Nick Hobson, Nov 14 2006
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STATUS
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approved
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A057208
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Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.
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+10
25
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5, 29, 1237, 32171803229, 829, 405565189, 14717, 39405395843265000967254638989319923697097319108505264560061, 282860648026692294583447078797184988636062145943222437, 53, 421, 13, 109, 4133, 6476791289161646286812333, 461, 34549, 453690033695798389561735541
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OFFSET
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1,1
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.
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LINKS
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EXAMPLE
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a(3) = 1237 = 8*154 + 5 is the smallest suitable prime divisor of (5*29)*5*29 + 4 = 21029 = 17*1237. (Although 17 is the smallest prime divisor, 17 is not congruent to 5 modulo 8.)
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MATHEMATICA
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a={5}; q=1;
For[n=2, n<=7, n++,
q=q*Last[a];
AppendTo[a, Min[Select[FactorInteger[4+q^2][[All, 1]], Mod[#, 8]==5 &]]];
];
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PROG
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(PARI) lista(nn) = {v = vector(nn); v[1] = 5; print1(v[1], ", "); for (n=2, nn, f = factor(4 + prod(k=1, n-1, v[k])^2); for (k=1, #f~, if (f[k, 1] % 8 == 5, v[n] = f[k, 1]; break); ); print1(v[n], ", "); ); } \\ Michel Marcus, Oct 27 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A051334
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Euclid-Mullin sequence (A000945) with initial value a(1)=8191 instead of a(1)=2.
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+10
12
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8191, 2, 3, 7, 53, 1399, 5, 19, 646843, 26945441, 109, 443, 90670999, 280460690293140589, 907, 16293787, 3655513, 499483, 131, 21067, 143797, 54540542259000816707816058313971443, 392963, 977, 11, 5021, 179, 439, 353, 34417238589462247, 1193114397863177, 13, 59, 31643, 79399, 73, 43, 16639867
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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a[1]=8191; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
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PROG
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(PARI) spf(n)=my(f=factor(n)[1, 1]); f;
first(m)={my(v=vector(m)); v[1]=8191; for(i=2, m, v[i]=spf(1+prod(j=1, i-1, v[j]))); v; } /* Anders Hellström, Aug 18 2015 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A051309
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Euclid-Mullin sequence (A000945) with initial value a(1)=11 instead of a(1)=2.
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+10
5
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11, 2, 23, 3, 7, 10627, 433, 17, 13, 10805892983887, 73, 6397, 19, 489407, 2753, 87491, 18618443, 5, 31, 113, 41, 10723, 35101153, 25243, 374399, 966011, 293821591198219762366057, 234947, 4729, 27953, 3256171, 331, 613, 67, 272646324430637, 34281113, 21050393332691947013, 61, 97
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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a[1]=11; a[n_] := First[ Flatten[ FactorInteger[ 1+Product[ a[ j ], {j, 1, n-1} ] ] ] ]; Array[a, 10]
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PROG
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(PARI) lpf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m)); v[1]=11; for(i=2, m, v[i]=lpf(1+prod(j=1, i-1, v[j]))); v;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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