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A370131 a(n) = A068346(A276086(n)), where A068346 is the second arithmetic derivative, and A276086 is the primorial base exp-function.

%I A370131 #10 Feb 11 2024 14:20:29
%S A370131 0,0,0,1,5,10,0,1,12,1,16,44,7,39,16,42,608,391,55,135,365,455,1120,
%T A370131 2990,800,1100,1400,5425,12575,21025,0,6,7,1,20,103,16,1,1,32,271,320,
%U A370131 24,78,572,459,1031,2887,635,1670,1155,3335,19540,22130,4225,7700,18675,28100,68900,155425,9,18,20,54,704,631,24,251
%N A370131 a(n) = A068346(A276086(n)), where A068346 is the second arithmetic derivative, and A276086 is the primorial base exp-function.
%H A370131 Antti Karttunen, <a href="/A370131/b370131.txt">Table of n, a(n) for n = 0..2559</a>
%H A370131 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A370131 a(n) = A068346(A276086(n)) = A003415(A327860(n)).
%o A370131 (PARI)
%o A370131 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A370131 A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
%o A370131 A370131(n) = A003415(A327860(n));
%Y A370131 Cf. A003415, A068346, A276086, A327860, A370130.
%Y A370131 Cf. A002110 (positions of 0's after the initial zero), A328233 (positions of 1's), A328240 (positions of primes), A369651 (= a(A143293(n-1)), for n >= 1).
%K A370131 nonn
%O A370131 0,5
%A A370131 _Antti Karttunen_, Feb 10 2024