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

Author

Leroy Quet, Oct 27 2000

Keywords

• nonn
• easy
• mult
• changed

Examples

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

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See sigma- function.

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.

Programs

Magma

[&*PrimeDivisors(n)*SumOfDivisors(n div &*PrimeDivisors(n)): n in [1..70]]; // Vincenzo Librandi, May 14 2015

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 *)

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

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