A347146 a(n) = Sum_{d|n} (d^d)', where ' is the arithmetic derivative.

0, 4, 27, 1028, 3125, 233311, 823543, 201327620, 2324522961, 70000003129, 285311670611, 142657607406431, 302875106592253, 100008061430845691, 3503151123046878152, 590295810358906979332, 827240261886336764177, 826274569581229613840149, 1978419655660313589123979

Offset: 1

Author

Wesley Ivan Hurt, Oct 16 2021

Keywords

• nonn

Comments

Inverse Möbius transform of (n^n)'. - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(4) = (1^1)' + (2^2)' + (4^4)' = 1' + 4' + 256' = 0 + 4 + 1024 = 1028.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..385

Crossrefs

Cf. A003415, A068327.

Programs

Maple

A347146 := proc(n)
    add(A068327(d),d=numtheory[divisors](n)) ;
end proc:
seq( A347146(n),n=1..10) ; # R. J. Mathar, Oct 19 2021

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#^#] &]; 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(d^d)); \\ Michel Marcus, Oct 18 2021

Formula

a(n) = Sum_{d|n} A068327(d). - R. J. Mathar, Oct 19 2021