%I #13 Apr 30 2022 22:51:33

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A353576/b353576.txt">Table of n, a(n) for n = 0..11550</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

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

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

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A351942(n) = A003415(A181819(n));

%o A353576(n) = A351942(A276086(n));

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

%Y Cf. A060735 (positions of 1's).

%Y 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.

%K nonn,base,look

%O 0,4

%A _Antti Karttunen_, Apr 30 2022