A306705 a(n) = Product_{d|n} d*tau(d), where tau(k) = the number of the divisors of k (A000005).

1, 4, 6, 48, 10, 576, 14, 1536, 162, 1600, 22, 497664, 26, 3136, 3600, 122880, 34, 1679616, 38, 2304000, 7056, 7744, 46, 3057647616, 750, 10816, 17496, 6322176, 58, 3317760000, 62, 23592960, 17424, 18496, 19600, 470184984576, 74, 23104, 24336, 23592960000, 82

Offset: 1

Author

Jaroslav Krizek, Mar 05 2019

Keywords

• nonn

Examples

			a(6) = 1*tau(1) * 2*tau(2) * 3*tau(3) * 6*tau(6) = (1*1) * (2*2) * (3*2) * (6*4) = 576.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A060640 (Sum_{d|n} d*tau(d)), A007955, A211776.

Programs

Magma

[&*[d * NumberOfDivisors(d): d in Divisors(n)]: n in [1..100]]

Maple

f:= proc(n) uses numtheory; local d;
  mul(d*tau(d),d = divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019

Mathematica

Table[n^(DivisorSigma[0, n]/2) * Product[DivisorSigma[0, k], {k, Divisors[n]}], {n, 1, 60}] (* Vaclav Kotesovec, Mar 10 2019 *)

PARI

a(n) = my(res = 1); fordiv(n, d, res *= d*numdiv(d)); res; \\ Michel Marcus, Mar 06 2019

Formula

a(p) = 2p for p = primes (A000040).

a(n) = (Product_{d|n} tau(d)) * (Product_{d|n} d) = A211776(n) * A007955(n).

From Robert Israel, Mar 24 2019: (Start)

a(p^k) = (k+1)! * p^(k*(k+1)/2) for primes p.

a(p*q) = 16*p^2*q^2 if p and q are distinct primes. (End)