A324986 a(n) = Sum_{d|n} (tau(d)*sigma(d)) where tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

1, 7, 9, 28, 13, 63, 17, 88, 48, 91, 25, 252, 29, 119, 117, 243, 37, 336, 41, 364, 153, 175, 49, 792, 106, 203, 208, 476, 61, 819, 65, 621, 225, 259, 221, 1344, 77, 287, 261, 1144, 85, 1071, 89, 700, 624, 343, 97, 2187, 188, 742, 333, 812, 109, 1456, 325, 1496

Offset: 1

Author

Jaroslav Krizek, Mar 23 2019

Keywords

• nonn

Comments

n divides a(n) for n: 1, 3, 4, 8, 12, 24, 28, 84, 88, 144, 264, 432, 440, 476, 1320, ...

Inverse Möbius transform of A064840. - Antti Karttunen, Mar 28 2019

Examples

			a(6) = tau(1)*sigma(1) + tau(2)*sigma(2) + tau(3)*sigma(3) + tau(6)*sigma(6) = (1*1) + (2*3) + (2*4) + (4*12) = 63.
		

Crossrefs

Cf. A000005, A000203, A064840, A324987.

Programs

Magma

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

Mathematica

Table[Sum[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Mar 23 2019 *)

PARI

a(n) = my(d=divisors(n)); sum(i=1, #d, numdiv(d[i])*sigma(d[i])) \\ Felix Fröhlich, Mar 23 2019

PARI

a(n) = sumdiv(n, d, numdiv(d)*sigma(d)); \\ Michel Marcus, Mar 24 2019

Formula

a(p) = 2p + 3 for p = primes (A000040).

a(n) = Sum_{d|n} A064840(d).