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A359603 Dirichlet inverse of function f(n) = 1+(A003415(n)*A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

%I A359603 #11 Jan 11 2023 20:59:58
%S A359603 1,-4,-7,-21,-19,30,-11,51,-132,-164,-91,-11,-51,-588,-935,-5904,-451,
%T A359603 -1402,-251,-5979,-7347,-13898,-2251,-25507,-12140,-27718,-99060,
%U A359603 -174307,-11251,11610,-15,52653,685,2410,-1095,24800,-71,-198,-2647,53673,-631,61020,-351,94173,-20052,-21368,-3151,207838
%N A359603 Dirichlet inverse of function f(n) = 1+(A003415(n)*A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.
%H A359603 Antti Karttunen, <a href="/A359603/b359603.txt">Table of n, a(n) for n = 1..11550</a>
%H A359603 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A359603 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A358669(n/d)) * a(d).
%o A359603 (PARI)
%o A359603 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A359603 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A359603 A358669(n) = (A003415(n)*A276086(n));
%o A359603 memoA359603 = Map();
%o A359603 A359603(n) = if(1==n,1,my(v); if(mapisdefined(memoA359603,n,&v), v, v = -sumdiv(n,d,if(d<n,(1+A358669(n/d))*A359603(d),0)); mapput(memoA359603,n,v); (v)));
%Y A359603 Cf. A003415, A276086, A358669, A359590 (parity of terms), A359604 [= a(n) mod 60].
%Y A359603 Cf. also A359427, A359589.
%K A359603 sign,easy
%O A359603 1,2
%A A359603 _Antti Karttunen_, Jan 11 2023