A348304 a(n) = Sum_{d|n} d'', where d'' is the second arithmetic derivative of d (A068346).

0, 0, 0, 4, 0, 1, 0, 20, 5, 1, 0, 37, 0, 6, 12, 100, 0, 16, 0, 49, 7, 1, 0, 101, 7, 8, 32, 90, 0, 15, 0, 276, 9, 1, 16, 144, 0, 10, 32, 137, 0, 15, 0, 117, 33, 10, 0, 421, 9, 47, 24, 104, 0, 151, 32, 202, 13, 1, 0, 191, 0, 14, 32, 916, 21, 12, 0, 161, 15, 24, 0, 428, 0, 16, 35

Offset: 1

Author

Wesley Ivan Hurt, Oct 10 2021

Keywords

• nonn
• look

Comments

Sum of the 2nd arithmetic derivatives of the divisors of n.

Examples

			a(8) = 20; a(8) = 1'' + 2'' + 4'' + 8'' = 0 + 0 + 4 + 16 = 20.
		

Links

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

Crossrefs

Inverse Möbius transform of A068346 (2nd arithmetic derivative).

Cf. A319684, A348279.

Programs

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, ad(ad(d))); \\ Michel Marcus, Oct 11 2021

Formula

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

a(n) = Sum_{d|n} A068346(d). - Antti Karttunen, Dec 07 2021