Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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