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

1,2

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.

Links

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

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 increasing sequence of divisors of n.

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

Example

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.

Prog

(Magma) [SumOfDivisors(n)*(Order(AlternatingGroup(NumberOfDivisors(n)+1))): n in [1..100]]
(PARI)
A001710(n) = if( n<2, n>=0, n!/2);
A246916(n) = (sigma(n) * A001710(numdiv(n) + 1)); \\ Antti Karttunen, Sep 10 2017

Crossrefs

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

Sequence in context: A140145 A166643 A038302 * A188001 A273614 A281224

Adjacent sequences: A246913 A246914 A246915 * A246917 A246918 A246919

Keyword

nonn

Author

Jaroslav Krizek, Sep 12 2014

Status

approved