A246916 - OEIS

%I #11 Sep 08 2022 08:46:09

%S 1,9,12,84,18,720,24,900,156,1080,36,70560,42,1440,1440,11160,54,

%T 98280,60,105840,1920,2160,72,10886400,372,2520,2400,141120,90,

%U 13063680,96,158760,2880,3240,2880,165110400,114,3600,3360,16329600,126,17418240,132,211680

%N Sum of the cumulative sums of all the permutations of divisors of number n.

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

%H Antti Karttunen, <a href="/A246916/b246916.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

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

%e 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.

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

%o (PARI)

%o A001710(n) = if( n<2, n>=0, n!/2);

%o A246916(n) = (sigma(n) * A001710(numdiv(n) + 1)); \\ _Antti Karttunen_, Sep 10 2017

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

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Sep 12 2014