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A057723 Sum of positive divisors of n that are divisible by every prime that divides n. 4
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 (list; graph; refs; listen; history; text; internal format)
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)). - 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

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 *)

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

Cf. A000203 (sigma), A007947 (rad).

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

Sequence in context: A323309 A322857 A051377 * A142151 A003968 A076734

Adjacent sequences:  A057720 A057721 A057722 * A057724 A057725 A057726

KEYWORD

nonn,mult

AUTHOR

Leroy Quet, Oct 27 2000

STATUS

approved

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)