A353576 Arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A003415(A181819(A276086(n))).

0, 1, 1, 4, 1, 5, 1, 4, 4, 12, 5, 16, 1, 5, 5, 16, 6, 21, 1, 7, 7, 24, 8, 31, 1, 9, 9, 32, 10, 41, 1, 4, 4, 12, 5, 16, 4, 12, 12, 32, 16, 44, 5, 16, 16, 44, 21, 60, 7, 24, 24, 68, 31, 92, 9, 32, 32, 92, 41, 124, 1, 5, 5, 16, 6, 21, 5, 16, 16, 44, 21, 60, 6, 21, 21, 60, 27, 81, 8, 31, 31, 92, 39, 123, 10, 41, 41, 124

Offset: 0

Author

Antti Karttunen, Apr 30 2022

Keywords

• nonn
• base
• look

Links

• Antti Karttunen, Table of n, a(n) for n = 0..11550
• Index entries for sequences related to primorial base

Crossrefs

Cf. A003415, A181819, A276086, A328835, A351942, A353524.

Cf. A060735 (positions of 1's).

Cf. also A353575 which has quite a similar scatter plot, while on the contrast, A353577 has a very different look, explained by the contribution of A353524.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351942(n) = A003415(A181819(n));
A353576(n) = A351942(A276086(n));

Formula

a(n) = A351942(A276086(n)) = A003415(A328835(n)) = A003415(A181819(A276086(n))).

a(n) = A353524(n) * A353577(n).