A325030 a(n) = Product_{d|n} (sigma(d)*pod(d)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

1, 6, 12, 336, 30, 31104, 56, 322560, 4212, 324000, 132, 84276412416, 182, 1580544, 1944000, 10239344640, 306, 2483164449792, 380, 6096384000000, 9483264, 13799808, 552, 1610547321930095001600, 116250, 31004064, 122821920, 108806975520768, 870

Offset

1,2

Comments

n divides a(n) for all n.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

a(n) = Product_{d|n} sigma(d) * Product_{d|n} pod(d) = A206032(n) * A266265(n).

a(p) = p*(p+1) for p = primes (A000040).

Example

a(6) = (sigma(1)*pod(1)) * (sigma(2)*pod(2)) * (sigma(3)*pod(3)) * (sigma(6)*pod(6)) = (1*1) * (3*2) * (4*3) * (12*36) = 31104.

Mathematica

Table[Times@@(DivisorSigma[1, #]Times@@Divisors[#]&/@Divisors[n]), {n, 30}] (* Harvey P. Dale, Dec 10 2024 *)

Prog

(Magma) [&*[&+ [c: c in Divisors(d)] * &*[c: c in Divisors(d)]: d in Divisors(n)]: n in [1..100]];
(PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, my(dd=divisors(d[k])); vecsum(dd)*vecprod(dd)); \\ Michel Marcus, Apr 25 2019

Crossrefs

Cf. A000203, A007955, A325029.

Sequence in context: A158046 A097174 A334809 * A191415 A259130 A032511

Adjacent sequences: A325027 A325028 A325029 * A325031 A325032 A325033

Keyword

nonn

Author

Jaroslav Krizek, Apr 25 2019

Status

approved