A294614 - OEIS

%I #24 Mar 28 2024 09:03:02

%S 0,0,1,0,0,0,0,2,0,2,0,2,3,0,0,0,3,4,0,0,0,0,8,4,3,0,3,6,0,0,5,0,7,4,

%T 0,0,0,18,0,0,0,0,9,4,12,4,0,14,0,0,5,8,11,0,0,6,0,12,9,0,5,0,13,6,5,

%U 10,7,14,0,0,5,0,31,0,5,0,7,30,0,12,0,0,17,6,0,0,13,18,9,8

%N Sum of the divisors of 12*n - 1, divided by 12, minus n.

%C a(n) = 0 iff n is in A138620.

%C First occurrence of k > -1: 1, 3, 8, 13, 18, 31, 28, 33, 23, 43, 66, 53, 45, 63, 48, 101, 166, etc.

%F a(n) = sigma(12*n-1)/12 - n = A000203(A017653(n-1))/12 - n.

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - _Amiram Eldar_, Mar 28 2024

%e a(13) = 3 since d(12*13-1)/12 - 13 = 192/12 - 13 = 16 - 13 = 3.

%t a[n_] := DivisorSigma[1, 12 n - 1]/12 - n; Array[a, 90]

%o (PARI) a(n) = sigma(12*n-1)/12 - n;

%Y Inspired by A291900.

%Y Cf. A000203, A017653, A068231, A086463, A138620.

%K nonn,easy

%O 1,8

%A _Omar E. Pol_ and _Robert G. Wilson v_, Nov 04 2017