A348393 a(n) = Sum_{d|n} (n^d)', where ' is the arithmetic derivative.

0, 5, 28, 1060, 3126, 233885, 823544, 201351372, 2324524398, 70000350147, 285311670612, 142657631177872, 302875106592254, 100008061904383173, 3503151123048905408, 590295810427425653792, 827240261886336764178, 826274569583310299739525, 1978419655660313589123980, 2516582400000122880019968984

Offset: 1

Author

Wesley Ivan Hurt, Oct 16 2021

Keywords

• nonn

Examples

			a(4) = 1060; a(4) = (4^1)' + (4^2)' + (4^4)' = 4' + 16' + 256' = 4 + 32 + 1024 = 1060.
		

Crossrefs

Cf. A003415, A125137.

Programs

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[n^#] &]; Array[a, 20] (* Amiram Eldar, Oct 16 2021 *)

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, ad(n^d)); \\ Michel Marcus, Oct 18 2021

Formula

If p is prime, a(p) = p^p + 1. See A125137. - Bernard Schott, Oct 18 2021