%I #18 Aug 18 2024 09:18:16

%S 1,2,2,6,2,4,2,7,11,4,2,9,2,4,4,23,2,14,2,9,4,4,2,11,27,4,12,9,2,8,2,

%T 24,4,4,4,55,2,4,4,11,2,8,2,9,14,4,2,28,51,30,4,9,2,16,4,11,4,4,2,15,

%U 2,4,14,88,4,8,2,9,4,8,2,58,2,4,30,9,4,8,2,28,93,4,2,15,4,4,4

%N a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of squares (A010052).

%C For each divisor d of n, add d if d is a square, otherwise add 1 [see example].

%F a(p) = 2 iff p is prime. - _Wesley Ivan Hurt_, Nov 28 2021

%F a(n) = A035316(n) + A056595(n). - _R. J. Mathar_, Aug 18 2024

%e The divisors of 12 are 1, 2, 3, 4, 6, and 12 with squares 1 and 4, so a(12) = 1 + 1 + 1 + 4 + 1 + 1 = 9 (respectively).

%t a[n_] := DivisorSum[n, If[IntegerQ @ Sqrt[#], #, 1] &]; Array[a, 100] (* _Amiram Eldar_, Nov 15 2021 *)

%o (PARI) a(n) = sumdiv(n, d, if (issquare(d), d, 1)); \\ _Michel Marcus_, Nov 15 2021

%o (PARI) a(n) = {my(f = factor(n), cf = f, res); cf[,2]\=2; res = numdiv(f)-prod(i = 1, #f~, cf[i, 2]+1); res+=prod(i = 1, #f~, ((f[i,1]^(2*(cf[i,2]+1))-1)/(f[i,1]^2-1))); res } \\ _David A. Corneth_, Nov 16 2021

%Y Cf. A010052, A035316.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Nov 15 2021