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

%I A246916 #23 Aug 08 2025 15:04:17
%S A246916 1,9,12,84,18,720,24,900,156,1080,36,70560,42,1440,1440,11160,54,
%T A246916 98280,60,105840,1920,2160,72,10886400,372,2520,2400,141120,90,
%U A246916 13063680,96,158760,2880,3240,2880,165110400,114,3600,3360,16329600,126,17418240,132,211680
%N A246916 Sum of the cumulative sums of all the permutations of divisors of number n.
%C A246916 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.
%C A246916 Sequences A064945 and A064944 are sequences of minimal and maximal values of cumulative sums of all the permutations of divisors of number n.
%H A246916 Antti Karttunen, <a href="/A246916/b246916.txt">Table of n, a(n) for n = 1..10000</a>
%F A246916 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.
%F A246916 a(n) = sigma(n) * A001710(tau(n) + 1) = A000203(n) * A001710(A000005(n)+1).
%e A246916 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.
%t A246916 A246916[n_] := DivisorSigma[1, n]*(DivisorSigma[0, n] + 1)!/2;
%t A246916 Array[A246916, 50] (* _Paolo Xausa_, Aug 08 2025 *)
%o A246916 (Magma) [SumOfDivisors(n)*(Order(AlternatingGroup(NumberOfDivisors(n)+1))): n in [1..100]];
%o A246916 (PARI)
%o A246916 A001710(n) = if( n<2, n>=0, n!/2);
%o A246916 A246916(n) = (sigma(n) * A001710(numdiv(n) + 1)); \\ _Antti Karttunen_, Sep 10 2017
%Y A246916 Cf. A000005, A000203, A001710, A064945, A064944, A130674.
%K A246916 nonn
%O A246916 1,2
%A A246916 _Jaroslav Krizek_, Sep 12 2014