A246916 Sum of the cumulative sums of all the permutations of divisors of number n.

1, 9, 12, 84, 18, 720, 24, 900, 156, 1080, 36, 70560, 42, 1440, 1440, 11160, 54, 98280, 60, 105840, 1920, 2160, 72, 10886400, 372, 2520, 2400, 141120, 90, 13063680, 96, 158760, 2880, 3240, 2880, 165110400, 114, 3600, 3360, 16329600, 126, 17418240, 132, 211680

Offset: 1

Author

Jaroslav Krizek, Sep 12 2014

Keywords

• nonn

Comments

For number n there are A130674(n) = tau(n)! = A000005(n)! permutations of divisors of number n and the same number of their cumulative sums. This sequence is sequence of sums of these sums.

Sequences A064945 and A064944 are sequences of minimal and maximal values of cumulative sums of all the permutations of divisors of number n.

Examples

			For n = 4; there are tau(4)! = 6 permutations of divisors of number 4: (1, 2, 4); (1, 4, 2); (2, 1, 4); (2, 4, 1); (4, 1, 2); (4, 2, 1). Sum of their cumulative sums = 11 + 13 + 12 + 15 + 16 + 17 = 84.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A001710, A064945, A064944, A130674.

Programs

Magma

[SumOfDivisors(n)*(Order(AlternatingGroup(NumberOfDivisors(n)+1))): n in [1..100]];

Mathematica

A246916[n_] := DivisorSigma[1, n]*(DivisorSigma[0, n] + 1)!/2;
Array[A246916, 50] (* Paolo Xausa, Aug 08 2025 *)

PARI

A001710(n) = if( n<2, n>=0, n!/2);
A246916(n) = (sigma(n) * A001710(numdiv(n) + 1)); \\ Antti Karttunen, Sep 10 2017

Formula

a(n) = A130674(n) * (A064945(n) + A064944(n)) / 2 = (tau(n))! * ((Sum_{i=1..tau(n)} (tau(n) - i + 1)*d_i) + (Sum_{i=1..tau(n)} i*d_i)) / 2; where {d_i}, i = 1..tau(n) is the increasing sequence of divisors of n.

a(n) = sigma(n) * A001710(tau(n) + 1) = A000203(n) * A001710(A000005(n)+1).