A350743 - OEIS

%I #27 Jan 18 2022 03:29:37

%S 1,1,2,1,3,3,2,3,1,5,4,5,4,5,6,1,5,4,3,7,4,5,5,8,1,7,6,6,8,9,3,5,8,7,

%T 9,3,4,10,6,13,7,8,4,8,9,7,9,5,3,5,10,10,7,13,8,14,8,12,10,18,3,10,7,

%U 1,14,10,5,16,11,12,5,12,6,9,10,10,8,14,5,11,2,13,13,15,14

%N Number of numbers k, 1 <= k <= n, such that k | sigma_k(n).

%F a(n) = n - Sum_{k=1..n} (ceiling(sigma_k(n)/k) - floor(sigma_k(n)/k)).

%e a(3) = 2; we have 1 | sigma_1(3) = 1 + 3 = 4 and 2 | sigma_2(3) = 1^2 + 3^2 = 10. (Note that 3 does not divide sigma_3(3) = 1^3 + 3^3 = 28.)

%t Table[n - Sum[Ceiling[DivisorSigma[k, n]/k] - Floor[DivisorSigma[k, n]/k], {k, n}], {n, 100}]

%o (PARI) a(n) = sum(k=1, n, (sigma(n,k) % k) == 0); \\ _Michel Marcus_, Jan 13 2022

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A350743(n):

%o     f = list(factorint(n).items())

%o     return sum(1 for k in range(1,n+1) if prod(p**((q+1)*k)-1 for p,q in f)//prod(p**k-1 for p,q in f) % k == 0) # _Chai Wah Wu_, Jan 17 2022

%Y Cf. A000203 (sigma), A348399.

%K nonn

%O 1,3

%A _Wesley Ivan Hurt_, Jan 13 2022