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Search: a157290 -id:a157290
Displaying 1-5 of 5 results found.
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A008966
a(n) = 1 if n is squarefree, otherwise 0.
+10
290
1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).
The infinite lower triangular matrix with A008966 on the main diagonal and the rest zeros is the square of triangle A143255. - Gary W. Adamson, Aug 02 2008
LINKS
Daniel Forgues, Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Moebius Function
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n
FORMULA
Dirichlet g.f.: zeta(s)/zeta(2s).
a(n) = abs(mu(n)), where mu is the Moebius function (A008683).
a(n) = 0^(bigomega(n) - omega(n)), where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller, Apr 05 2003
Multiplicative with p^e -> 0^(e - 1), p prime and e > 0. - Reinhard Zumkeller, Jul 15 2003
a(n) = 0^(A046951(n) - 1). - Reinhard Zumkeller, May 20 2007
a(n) = 1 - A107078(n). - Reinhard Zumkeller, Oct 03 2008
a(n) = floor(rad(n)/n), where rad() is A007947. - Enrique Pérez Herrero, Nov 13 2009
A175046(n) = a(n)*A073311(n). - Reinhard Zumkeller, Apr 05 2010
a(n) = floor(A000005(n^2)/A007425(n)). - Enrique Pérez Herrero, Apr 15 2010
a(A005117(n)) = 1; a(A013929(n)) = 0; a(n) = A013928(n + 1) - A013928(n). - Reinhard Zumkeller, Jul 05 2010
a(n) * A112526(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = mu(n) * lambda(n) = A008836(n) * A008683(n). - Enrique Pérez Herrero, Nov 29 2013
a(n) = Sum_{d|n} 2^omega(d)*mu(n/d). - Geoffrey Critzer, Feb 22 2015
a(n) = A085357(A156552(n)). - Antti Karttunen, Mar 06 2017
Limit_{n->oo} (1/n)*Sum_{j=1..n} a(j) = 6/Pi^2. - Andres Cicuttin, Aug 13 2017
a(1) = 1; a(n) = -Sum_{d|n, d < n} (-1)^bigomega(n/d) * a(d). - Ilya Gutkovskiy, Mar 10 2021
MAPLE
A008966 := proc(n) if numtheory[issqrfree](n) then 1 ; else 0 ; end if; end proc: # R. J. Mathar, Mar 14 2011
MATHEMATICA
A008966[n_] := Abs[MoebiusMu[n]]; Table[A008966[n], {n, 100}] (* Enrique Pérez Herrero, Apr 15 2010 *)
Table[If[SquareFreeQ[n], 1, 0], {n, 100}] (* or *) Boole[SquareFreeQ/@ Range[ 100]] (* Harvey P. Dale, Feb 28 2015 *)
PROG
(MuPAD) func(abs(numlib::moebius(n)), n):
(Magma) [ Abs(MoebiusMu(n)) : n in [1..100]];
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1+X))[n]
(PARI) a(n)=issquarefree(n) \\ Michel Marcus, Feb 22 2015
(Haskell)
a008966 = abs . a008683
-- Reinhard Zumkeller, Dec 13 2015, Dec 15 2014, May 27 2012, Jan 25 2012
(Python)
from sympy import factorint
def A008966(n): return int(max(factorint(n).values(), default=1)==1) # Chai Wah Wu, Apr 05 2023
CROSSREFS
Cf. A005117, A008836 (Dirichlet inverse), A013928 (partial sums).
Cf. A179211, A179212, A179213, A179214, A179215.
Cf. A020639, A073576, A087188.
Cf. A124010, A212793, A085357, A156552.
Parity of A002033.
Cf. A047999, A036987
Cf. A082020 (Dgf at s=2), A157289 (Dgf at s=3), A157290 (Dgf at s=4).
KEYWORD
easy,nonn,mult
AUTHOR
N. J. A. Sloane
EXTENSIONS
Deleted an unclear comment. - N. J. A. Sloane, May 30 2021
STATUS
approved
A030514
a(n) = prime(n)^4.
+10
112
16, 81, 625, 2401, 14641, 28561, 83521, 130321, 279841, 707281, 923521, 1874161, 2825761, 3418801, 4879681, 7890481, 12117361, 13845841, 20151121, 25411681, 28398241, 38950081, 47458321, 62742241, 88529281, 104060401, 112550881, 131079601, 141158161
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers with 5 divisors (1, p, p^2, p^3, p^4, where p is the n-th prime). - Alexandre Wajnberg, Jan 15 2006
Subsequence of A036967. - Reinhard Zumkeller, Feb 05 2008
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
The general product formula for even s is: product_{p = A000040} (p^s-1)/(p^s+1) = 2*Bernoulli(2s)/( binomial(2s, s)*Bernoulli^2(s)), where the infinite product is over all primes. Here, with s = 4, product_{n = 1, 2, ...} (a(n)-1)/(a(n)+1) = 6/7. In A030516, where s = 6, the product of the ratios is 691/715. For s = 8, the 8th row in A120458, the corresponding product of ratios is 7234/7293. - R. J. Mathar, Feb 01 2009
Except for the first three terms, all others are congruent to 1 mod 240. - Robert Israel, Aug 29 2014
LINKS
R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature
FORMULA
a(n) = A000040(n)^(5-1) = A000040(n)^4, where 5 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 5. - Alexandre Wajnberg, Jan 15 2006
A056595(a(n)) = 2. - Reinhard Zumkeller, Aug 15 2011
Sum_{n>=1} 1/a(n) = P(4) = 0.0769931397... (A085964). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(4)/zeta(8) = 105/Pi^4 (A157290).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(4) = 90/Pi^4 (A215267). (End)
MAPLE
map(p -> p^4, select(isprime, [2, seq(2*i+1, i=1..100)])); # Robert Israel, Aug 29 2014
MATHEMATICA
Array[Prime[#]^4 &, 5!] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
PROG
(Sage)
[p**4 for p in prime_range(100)]
# Zerinvary Lajos, May 15 2007
(Magma) [NthPrime(n)^4: n in [1..100] ]; // Vincenzo Librandi, Apr 22 2011
(PARI) a(n)=prime(n)^4 \\ Charles R Greathouse IV, Mar 21 2013
(Haskell)
a030514 = (^ 4) . a000040
a030514_list = map (^ 4) a000040_list
-- Reinhard Zumkeller, Jun 03 2015
CROSSREFS
Cf. A030078, A085964, A131991, A131992, A000005, A000040, A001248, A157290, A215267.
Cf. A258601.
KEYWORD
nonn,easy
AUTHOR
Jeff Burch
EXTENSIONS
Description corrected by Eric W. Weisstein
STATUS
approved
A082020
Decimal expansion of 15/Pi^2.
+10
39
1, 5, 1, 9, 8, 1, 7, 7, 5, 4, 6, 3, 5, 0, 6, 6, 5, 7, 1, 6, 5, 8, 1, 9, 1, 9, 4, 8, 1, 4, 5, 9, 1, 4, 5, 8, 3, 5, 6, 5, 3, 8, 1, 6, 2, 0, 0, 8, 3, 6, 9, 8, 2, 3, 2, 6, 8, 4, 1, 3, 5, 4, 7, 8, 4, 1, 2, 5, 9, 6, 8, 1, 4, 4, 3, 3, 5, 3, 1, 6, 1, 7, 8, 6, 8, 1, 3, 9, 1, 0, 8, 8, 8, 4, 3, 2, 7, 5, 6
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
3/(2*Pi^2) (the same decimal expansion with an offset 0) is the probability that the greatest common divisor of two numbers selected at random is 2 (Christopher, 1956). - Amiram Eldar, May 23 2020
Sum of 1/n^2 over all squarefree n, see Penn link. - Charles R Greathouse IV, Jan 01 2022
Equals the asymptotic mean of the abundancy index of the cubefree numbers (A004709) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
Werner Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Vol. 14, No. 2 (2015), pp. 73-88.
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Michael Penn, Irregular numbers and the coolest sum I have ever seen!, 2021 video.
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link (p. 230).
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Prime Products.
Index entries for transcendental numbers.
FORMULA
Product_{n >= 1} (1+1/prime(n)^2) = 15/Pi^2 (Ramanujan).
Equals zeta(2)/zeta(4) = A013661/A013662 = Sum_{n>=1} mu(n)^2/n^2 = Sum_{n>=1} |mu(n)|/n^2 . - Enrique Pérez Herrero, Jan 15 2012
Equals Sum_{n>=1} 1/A005117(n)^2 . - Enrique Pérez Herrero, Mar 30 2012
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} psi(k)/k, where psi(k) is the Dedekind psi function (A001615). - Amiram Eldar, May 12 2019.
Equals Sum_{k>=1} A007434(k)/k^4. - Amiram Eldar, Jan 25 2024
EXAMPLE
1.51981775463506657...
MAPLE
evalf(15/Pi^2, 120); # G. C. Greubel, Oct 18 2019
MATHEMATICA
A082020[digits_] := First[RealDigits[Zeta[2]/Zeta[4], 10, digits]]; A082020[100] (* Enrique Pérez Herrero, Jan 15 2012 *)
RealDigits[15/Pi^2, 10, 120][[1]] (* Harvey P. Dale, Jun 23 2019 *)
PROG
(PARI) 15/Pi^2 \\ Michel Marcus, Oct 18 2019
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 15/Pi(R)^2; // G. C. Greubel, Oct 18 2019
(Sage) numerical_approx(15/pi^2, digits=100) # G. C. Greubel, Oct 18 2019
CROSSREFS
Cf. A001615, A004709, A005117, A007434, A013661, A013662, A157290.
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, May 09 2003
STATUS
approved
A157294
Decimal expansion of 1575/Pi^6.
+10
2
1, 6, 3, 8, 2, 5, 4, 3, 2, 0, 4, 4, 0, 9, 6, 7, 3, 6, 6, 3, 4, 1, 4, 9, 4, 2, 7, 4, 9, 8, 9, 8, 7, 3, 5, 5, 4, 9, 1, 8, 7, 0, 2, 5, 2, 6, 6, 4, 4, 3, 4, 4, 7, 1, 8, 0, 7, 2, 9, 0, 0, 6, 7, 4, 8, 9, 2, 5, 0, 4, 2, 3, 5, 5, 7, 4, 4, 7, 9, 0, 4, 1, 3, 4, 8, 3, 1, 5, 9, 2, 4, 6, 3, 0, 4, 9, 2, 3, 6, 9, 2, 5, 6, 9, 1
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Equals the asymptotic mean of the abundancy index of the 7-free numbers (numbers that are not divisible by a 7th power other than 1) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023
LINKS
Table of n, a(n) for n=1..105.
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Index entries for transcendental numbers.
FORMULA
Equals Product_{p = primes = A000040} (1+1/p^2+1/p^4+1/p^6).
Equals A013661/A013666 = A082020*A157290 = Product_{i>=1} (1+1/A001248(i)+1/A030514(i)+1/A030516(i)) = 1575*A092746.
EXAMPLE
1.63825432044096736634149427498... = (1+1/2^2+1/2^4+1/2^6)*(1+1/3^2+1/3^4+1/3^6)*(1+1/5^2+1/5^4+1/5^6)*(1+1/7^2+1/7^4+1/7^6)*...
MAPLE
evalf(1575/Pi^6) ;
MATHEMATICA
RealDigits[1575/Pi^6, 10, 120][[1]] (* Harvey P. Dale, May 26 2019 *)
PROG
(PARI) 1575/Pi^6 \\ Charles R Greathouse IV, Oct 01 2022
CROSSREFS
Cf. A001248, A030514, A030516, A082020, A013661, A013666, A092746, A157290.
KEYWORD
cons,easy,nonn
AUTHOR
R. J. Mathar, Feb 26 2009
STATUS
approved
A269404
Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).
+10
2
1, 0, 1, 7, 0, 9, 2, 7, 6, 9, 1, 3, 0, 4, 9, 9, 2, 7, 6, 6, 4, 3, 2, 7, 2, 1, 3, 3, 0, 9, 7, 9, 0, 9, 9, 2, 0, 4, 9, 2, 2, 1, 9, 0, 7, 9, 4, 9, 4, 1, 0, 1, 1, 3, 4, 6, 6, 4, 6, 5, 1, 7, 9, 3, 8, 1, 8, 9, 3, 5, 3, 3, 5, 8, 3, 4, 2, 2, 7, 9, 4, 3, 1, 8, 1, 5, 1, 5, 9, 6, 4, 7, 8, 5, 0, 6, 6, 8, 9, 7, 8, 4, 5, 4, 6, 5, 1, 0, 6, 4, 0, 2, 6, 1, 3, 3, 6, 9, 3, 0
(list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
More generally, Product_{k >= 1} (1 + 1/prime(k)^m) = zeta(m)/zeta(2*m), where zeta(m) is the Riemann zeta function.
LINKS
Table of n, a(n) for n=1..120.
Eric Weisstein's World of Mathematics, Prime Products.
FORMULA
Equals zeta(6)/zeta(12).
Equals 675675/(691*Pi^6).
Equals Sum_{k>=1} 1/A005117(k)^6 = 1 + Sum_{k>=1} 1/A113851(k). - Amiram Eldar, Jun 27 2020
EXAMPLE
1.0170927691304992766432721330979099204922190794941...
MATHEMATICA
RealDigits[Zeta[6]/Zeta[12], 10, 120][[1]]
RealDigits[675675/(691 Pi^6), 10, 120][[1]]
PROG
(PARI) zeta(6)/zeta(12) \\ Amiram Eldar, Jun 11 2023
CROSSREFS
Cf. A005117, A082020, A113851, A157289, A157290, A157291.
KEYWORD
nonn,cons
AUTHOR
Ilya Gutkovskiy, Feb 25 2016
STATUS
approved
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