Displaying 1-10 of 11 results found.
1, 2, 6, 4, 30, 4, 210, 2, 12, 4, 2310, 2, 30030, 4, 36, 8, 510510, 2, 9699690, 2, 36, 4, 223092870, 12, 60, 4, 18, 2, 6469693230, 2, 200560490130, 2, 12, 4, 60, 16, 7420738134810, 4, 12, 12, 304250263527210, 2, 13082761331670030, 2, 6, 4, 614889782588491410, 2, 420, 2, 36, 2, 32589158477190044730, 4, 180, 24, 36, 4
PROG
(PARI) A373985(n) = { my(f=factor(n), m=1, s=0); for(i=1, #f~, my(x=prod(i=1, primepi(f[i, 1]), prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); gcd(m, s); };
0, 1, 1, 1, 1, 2, 1, 3, 1, 8, 1, 5, 1, 53, 1, 1, 1, 7, 1, 17, 6, 578, 1, 1, 1, 7508, 1, 107, 1, 19, 1, 5, 193, 127628, 4, 1, 1, 2424923, 2503, 3, 1, 109, 1, 1157, 7, 55773218, 1, 7, 1, 31, 14181, 15017, 1, 5, 13, 9, 269436, 1617423308, 1, 1, 1, 50140122533, 37, 3, 167, 1159, 1, 255257, 18591073, 121, 1, 1, 1, 1855184533703
PROG
(PARI) A373986(n) = { my(f=factor(n), m=1, s=0); for(i=1, #f~, my(x=prod(i=1, primepi(f[i, 1]), prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); s/gcd(m, s); };
1, 1, 1, 1, 1, 3, 1, 4, 3, 15, 1, 12, 1, 105, 5, 2, 1, 36, 1, 60, 35, 1155, 1, 4, 15, 15015, 12, 420, 1, 180, 1, 16, 1155, 255255, 105, 9, 1, 4849845, 15015, 20, 1, 1260, 1, 4620, 180, 111546435, 1, 48, 105, 900, 85085, 60060, 1, 108, 385, 70, 1616615, 3234846615, 1, 18, 1, 100280245065, 1260, 16, 5005, 13860, 1, 1021020
PROG
(PARI) A373987(n) = { my(f=factor(n), m=1, s=0); for(i=1, #f~, my(x=prod(i=1, primepi(f[i, 1]), prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); m/gcd(m, s); };
1, 0, 0, 0, 0, 4, 0, 2, 24, 28, 0, 14, 0, 208, 144, 8, 0, 58, 0, 86, 1044, 2308, 0, 36, 840, 30028, 198, 626, 0, 322, 0, 22, 11544, 510508, 6060, 128, 0, 9699688, 150144, 204, 0, 2302, 0, 6926, 1038, 223092868, 0, 82, 43680, 1738, 2552544, 90086, 0, 412, 66960, 1464, 48498444, 6469693228, 0, 680, 0, 200560490128
PROG
(PARI) A373984(n) = { my(f=factor(n), m=1, s=0); for(i=1, #f~, my(x=prod(i=1, primepi(f[i, 1]), prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); (m-s); };
1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 6, 4, 2, 2, 1, 4, 1, 2, 1, 2, 6, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 6, 1, 1, 1, 4, 2, 1, 4, 2, 2, 2, 2, 1, 2, 6, 1, 2, 2, 2, 4, 1, 1, 5, 4, 1, 1, 1, 2, 1
PROG
(PARI)
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A373985(n) = { my(f=factor(n), m=1, s=0); for(i=1, #f~, my(x=prod(i=1, primepi(f[i, 1]), prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); gcd(m, s); };
Integer log of n: sum of primes dividing n (with repetition). Also called sopfr(n).
(Formerly M0461 N0168)
+10
678
0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, 8, 17, 8, 19, 9, 10, 13, 23, 9, 10, 15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10, 37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14, 12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 12, 18, 16, 67, 21, 26, 14, 71, 12, 73, 39, 13, 23, 18, 18
COMMENTS
MacMahon calls this the potency of n.
Downgrades the operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopfr(40) = 2 * 3 + 5 = 11.
Consider all ways of writing n as a product of zero, one, or more factors; sequence gives smallest sum of terms. - Amarnath Murthy, Jul 07 2001 a(n) <= n for all n, and a(n) = n iff n is 4 or a prime.
Look at the graph of this sequence. At the lower edge of the logarithmic scatterplot there is a set of fuzzy but unmistakable diagonal fringes, sloping toward the southeast. Their spacing gradually increases, and their slopes gradually decrease; they are more distinct toward the lower edge of the range. Is any explanation known? - Allan C. Wechsler, Oct 11 2015 For n >= 2, the glb and lub are: 3 * log(n) / log(3) <= a(n) <= n, where the lub occurs when n = 3^k, k >= 1. (Jakimczuk 2012) - Daniel Forgues, Oct 12 2015 Differs from A337310 beginning with n at 64, 192, 256, 320, 448, 512, ..., . The number of terms which equal k is A000607(k). The first occurrence of k>1 is A056240(k). The last occurrence of k>1 is A000792(k). The Amarnath Murthy comment of Jul 07 2001 is a result of the fundamental theorem of arithmetic. (End)
REFERENCES
K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Amarnath Murthy, Generalization of Partition function and introducing Smarandache Factor Partition, 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 1.4.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 89.
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).
LINKS
Steve Witham, Linear-log plot (The clear upper lines are n (the primes), n/2, n/3, n/4... but there is a dark band at sqrt(n).) Steve Witham, Log-log plot (Differently interesting at the lower edge. Higher up, you can see sqrt(n), sqrt(n)/2, maybe sqrt(n)/3.)
FORMULA
If n = Product p_j^k_j then a(n) = Sum p_j * k_j.
Dirichlet g.f. f(s)*zeta(s), where f(s) = Sum_{p prime} p/(p^s-1) = Sum_{k>0} primezeta(k*s-1) is the Dirichlet g.f. for A120007. Totally additive with a(p^e) = p*e. - Franklin T. Adams-Watters, Jun 02 2006 Sum_{n>=1} (-1)^a(n)/n^s = ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s), if Re(s)>1 and 0 if s=1 (Alladi and Erdős, 1977). - Amiram Eldar, Nov 02 2020
EXAMPLE
a(24) = 2+2+2+3 = 9.
a(30) = 10: 30 can be written as 30, 15*2, 10*3, 6*5, 5*3*2. The corresponding sums are 30, 17, 13, 11, 10. Among these 10 is the least.
MAPLE
A001414 := proc(n) add( op(1, i)*op(2, i), i=ifactors(n)[2]) ; end proc:
MATHEMATICA
a[n_] := Plus @@ Times @@@ FactorInteger@ n; a[1] = 0; Array[a, 78] (* Ray Chandler, Nov 12 2005 *)
PROG
(PARI) a(n)=local(f); if(n<1, 0, f=factor(n); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]))
(Haskell)
a001414 1 = 0
a001414 n = sum $ a027746_row n
(Sage) [sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0, len(factor(n)))) for n in range(1, 79)] # Giuseppe Coppoletta, Jan 19 2015 (Python)
from sympy import factorint
return sum(p*e for p, e in factorint(n).items()) # Chai Wah Wu, Jan 08 2016 (Magma) [n eq 1 select 0 else (&+[j[1]*j[2]: j in Factorization(n)]): n in [1..100]]; // G. C. Greubel, Jan 10 2019
CROSSREFS
For sum of squares of prime factors see A067666, for cubes see A224787. Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A056239 (with a(p)=primepi(p)), A059975 (with a(p)=p-1), A064097 (with a(p)=1+a(p-1)), A113177 (with a(p)=Fib(p)), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#). For other completely additive sequences see the cross-references in A104244.
Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).
+10
147
0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10
COMMENTS
This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a( A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022
FORMULA
a(1) = 0, a(n) = (e1* A002110(i1-1) + ... + ez* A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez. Other identities.
For all n >= 0:
a( A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums] When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation: In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+ A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086: The sum or difference of the rhs-sequences is A108951: Here the two sequences are inverse permutations of each other:
Other correspondences:
a( A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation] (End)
MATHEMATICA
nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)
PROG
(Scheme, with memoization-macro definec)
(PARI) A276085(n) = { my(f=factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1, primepi(f[k, 1]-1), prime(i))); }; \\ Antti Karttunen, Mar 15 2021, Jun 23 2024 (Python)
from sympy import primorial, primepi, factorint
def a002110(n):
return 1 if n<1 else primorial(n)
def a(n):
f=factorint(n)
return sum(f[i]*a002110(primepi(i) - 1) for i in f)
CROSSREFS
Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576. Cf. A373145 [= gcd( A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd( A001414(n), a(n))], A373485 [= gcd( A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)]. Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
EXTENSIONS
Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024
For n > 1, a(n) is the least number of prime factors (counted with multiplicity) of any integer with n divisors; fully additive with a(p) = p-1.
+10
27
0, 1, 2, 2, 4, 3, 6, 3, 4, 5, 10, 4, 12, 7, 6, 4, 16, 5, 18, 6, 8, 11, 22, 5, 8, 13, 6, 8, 28, 7, 30, 5, 12, 17, 10, 6, 36, 19, 14, 7, 40, 9, 42, 12, 8, 23, 46, 6, 12, 9, 18, 14, 52, 7, 14, 9, 20, 29, 58, 8, 60, 31, 10, 6, 16, 13, 66, 18, 24, 11, 70, 7, 72, 37, 10, 20, 16, 15, 78, 8, 8, 41
COMMENTS
n*a(n) is the number of complex multiplications needed for the fast Fourier transform of n numbers, writing n = r1 * r2 where r1 is a prime.
This sequence with offset 1 and a(1) = 0 is completely additive with a(p^e) = e*(p-1) for prime p and e >= 0. - Werner Schulte, Feb 23 2019
REFERENCES
Herbert S. Wilf, Algorithms and complexity, Internet Edition, Summer, 1994, p. 56.
FORMULA
a(n) = Sum ( e_i * (p_i - 1) ) where n = Product ( p_i^e_i ) is the canonical factorization of n.
EXAMPLE
a(18) = 5 since 18 = 2*3^2, a(18) = 1*(2-1) + 2*(3-1) = 5.
MAPLE
local a, pf, p, e ;
a := 0 ;
for pf in ifactors(n)[2] do
p := op(1, pf) ;
e := op(2, pf) ;
a := a+e*(p-1) ;
end do:
a ;
MATHEMATICA
Table[Total[(First /@ FactorInteger[n] - 1) Last /@ FactorInteger[n]], {n, 1, 100}] (* Danny Marmer, Nov 13 2014 *) f[p_, e_] := e*(p - 1); a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 27 2023 *)
PROG
(PARI) a(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); } \\ Amiram Eldar, Mar 27 2023
CROSSREFS
Essentially same as A087656 apart from offset. Cf. A003159 (positions of even terms), A096268 (with offset 1, parity of terms), A373385 (positions of multiples of 3). Leftmost column of irregular table A355029. Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A276085 (with a(p)=p#/p), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).
AUTHOR
Yong Kong (ykong(AT)curagen.com), Mar 05 2001
a(1)=p_1, a(2)=p_1 + p_1*p_2, a(3)=p_1 + p_1*p_2 + p_1*p_2*p_3, ... where p_i is the i-th prime.
+10
10
2, 8, 38, 248, 2558, 32588, 543098, 10242788, 233335658, 6703028888, 207263519018, 7628001653828, 311878265181038, 13394639596851068, 628284422185342478, 33217442899375387208, 1955977793053588026278
COMMENTS
The only values of n for which a(n)/2 is prime are: 3, 5, 7, 11, 15, 47, 49. The corresponding 7 primes are: 19, 1279, 271549, 103631759509, 314142211092671239, 826811434211869939736645732264127163964562391958563586838409421490271014424018927729 , 41839936239750050346953677118447851613901200239299781782205558511980130628486398081201749. - Amiram Eldar, May 04 2017
EXAMPLE
a(4) = 248 because p_1 + p_1*p_2 + p_1*p_2*p_3 + p_1*p_2*p_3*p_4 = 2 + 6 + 30 + 210 = 248.
MAPLE
for n from 1 to 30 do printf(`%d, `, sum(product(ithprime(i), i=1..j), j=1..n)) od:
MATHEMATICA
Accumulate[Denominator[Accumulate[1/Prime[Range[20]]]]] (* Alonso del Arte, Mar 21 2013 *)
Lexicographically earliest infinite sequence such that a(i) = a(j) => A373986(i) = A373986(j), for all i, j >= 1.
+10
2
1, 2, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 2, 2, 8, 2, 9, 10, 11, 2, 2, 2, 12, 2, 13, 2, 14, 2, 6, 15, 16, 17, 2, 2, 18, 19, 4, 2, 20, 2, 21, 8, 22, 2, 8, 2, 23, 24, 25, 2, 6, 26, 27, 28, 29, 2, 2, 2, 30, 31, 4, 32, 33, 2, 34, 35, 36, 2, 2, 2, 37, 38, 39, 3, 40, 2, 14, 2, 41, 2, 38, 42, 43, 44, 15, 2, 38, 45, 46, 47, 48, 49, 2, 2, 50, 51, 17, 2, 52, 2, 19
COMMENTS
Restricted growth sequence transform of A373986.
PROG
(PARI)
up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
A373986(n) = { my(f=factor(n), m=1, s=0); for(i=1, #f~, my(x=prod(i=1, primepi(f[i, 1]), prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); s/gcd(m, s); };
v373988 = rgs_transform(vector(up_to, n, A373986(n)));
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