A348279 a(n) = Sum_{d|n} d*d', where d' is the arithmetic derivative of d (A003415).

0, 2, 3, 18, 5, 35, 7, 114, 57, 77, 11, 243, 13, 135, 128, 626, 17, 467, 19, 573, 220, 299, 23, 1395, 255, 405, 786, 1047, 29, 1160, 31, 3186, 476, 665, 432, 2835, 37, 819, 640, 3389, 41, 2100, 43, 2427, 1937, 1175, 47, 7283, 693, 2577, 1040, 3333, 53, 5570, 896, 6295, 1276

Offset: 1

Author

Wesley Ivan Hurt, Oct 09 2021

Keywords

• nonn

Examples

			a(4) = 18; a(4) = 1*1' + 2*2' + 4*4' = 1*0 + 2*1 + 4*4 = 18.
		

Links

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

Crossrefs

Cf. A003415 (arithmetic derivative).

Inverse Möbius transform of A190116.

Cf. also A347130.

Programs

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, d*ad(d)); \\ Michel Marcus, Oct 10 2021

Formula

a(p) = p for primes p since we have a(p) = 1*1' + p*p' = 1*0 + p*1 = p.

a(n) = Sum_{d|n} A190116(d). - Antti Karttunen, Dec 07 2021