A057723 Sum of positive divisors of n that are divisible by every prime that divides n.

1, 2, 3, 6, 5, 6, 7, 14, 12, 10, 11, 18, 13, 14, 15, 30, 17, 24, 19, 30, 21, 22, 23, 42, 30, 26, 39, 42, 29, 30, 31, 62, 33, 34, 35, 72, 37, 38, 39, 70, 41, 42, 43, 66, 60, 46, 47, 90, 56, 60, 51, 78, 53, 78, 55, 98, 57, 58, 59, 90, 61, 62, 84, 126, 65, 66, 67, 102, 69, 70

Offset

1,2

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Formula

If n = Product p_i^e_i then a(n) = Product (p_i + p_i^2 + ... + p_i^e_i).

a(n) = rad(n)*sigma(n/rad(n)) = A007947(n)*A000203(A003557(n)). - Ivan Neretin, May 13 2015

Dirichlet g.f.: zeta(s) * zeta(s-1) * Product(p prime, 1 - p^(-s) + p^(1-2*s)). - Robert Israel, May 13 2015

Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A330596 = Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.7485352596823635646442150486379106016416403430053244045... - Vaclav Kotesovec, Dec 18 2019

a(n) = Sum_{d|n, rad(d)=rad(n)} d. - R. J. Mathar, Jun 02 2020

Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Jun 10 2020

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * sigma(d). - Ilya Gutkovskiy, Apr 15 2021

Example

The divisors of 12 that are divisible by both 2 and 3 are 6 and 12. So a(12) = 6 + 12 = 18.

Maple

seq(mul(f[1]*(f[1]^f[2]-1)/(f[1]-1), f = ifactors(n)[2]), n = 1 .. 100); # Robert Israel, May 13 2015

Mathematica

Table[(b = Times @@ FactorInteger[n][[All, 1]])*DivisorSigma[1, n/b], {n, 70}] (* Ivan Neretin, May 13 2015 *)
f[p_, e_] := (p^(e+1)-1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

Prog

(Magma) [&*PrimeDivisors(n)*SumOfDivisors(n div &*PrimeDivisors(n)): n in [1..70]]; // Vincenzo Librandi, May 14 2015
(PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, f[i, 2]=1); my(pp = factorback(f)); sumdiv(n, d, if (! (d % pp), d, 0)); } \\ Michel Marcus, May 14 2015

Crossrefs

Row sums of triangle A284318.

Cf. A000203 (sigma), A007947 (rad), A005361 (number of these divisors).

Cf. A049060 and A060640 (other sigma-like functions).

Cf. A065487, A330596.

Sequence in context: A336465 A340774 A345068 * A335835 A361480 A142151

Adjacent sequences: A057720 A057721 A057722 * A057724 A057725 A057726

Keyword

nonn,easy,mult

Author

Leroy Quet, Oct 27 2000

Status

approved