A349330 a(n) = Sum_{d|n} d^c(d), where c is the characteristic function of squares (A010052).

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, 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, 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

Offset: 1

Author

Wesley Ivan Hurt, Nov 15 2021

Keywords

• nonn

Comments

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

Inverse Möbius transform of n^c(n), where c = A010052. - Wesley Ivan Hurt, Mar 31 2025

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A010052, A035316.

Programs

Mathematica

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

PARI

a(n) = sumdiv(n, d, if (issquare(d), d, 1)); \\ Michel Marcus, Nov 15 2021

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

Formula

a(p) = 2 iff p is prime. - Wesley Ivan Hurt, Nov 28 2021

a(n) = A035316(n) + A056595(n). - R. J. Mathar, Aug 18 2024