A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

0, 0, 2, 6, 8, 30, 18, 210, 24, 36, 70, 2310, 48, 30030, 434, 120, 64, 510510, 90, 9699690, 160, 672, 4642, 223092870, 120, 300, 60086, 162, 896, 6469693230, 270, 200560490130, 160, 6996, 1021054, 1260, 216, 7420738134810, 19399418, 90168, 360, 304250263527210, 1386, 13082761331670030, 9328, 450, 446185786, 614889782588491410, 288, 2940, 650, 1531632

Offset: 0

Author

Antti Karttunen, Oct 28 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..1001
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A002110, A034386, A108951, A328772.

Cf. also A003415, A258851, A328768, A328845, A328846.

Programs

PARI

A034386(n) = factorback(primes(primepi(n)));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A034386(f[i,1])/f[i, 1]));

PARI

A002110(n) = prod(i=1,n,prime(i));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1]))/f[i, 1]));

Formula

a(n) = n * Sum e_j * (p_j)#/p_j for n = Product p_j^e_j with (p_j)# = A034386(p_j).

A276150(a(n)) = A328772(n).