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A356304 The least k >= 0 such that A003415(n) and A276086(n+k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

%I A356304 #11 Nov 04 2022 11:26:19
%S A356304 0,0,0,0,24,0,4,3,0,0,0,0,4,1,0,0,0,0,4,9,0,0,0,5,4,3,0,0,0,0,0,177,0,
%T A356304 1,24,0,172,1,0,0,0,0,4,3,14,0,162,161,10,9,158,0,0,1,0,1,0,0,0,0,4,3,
%U A356304 2,1,0,0,4,1,0,0,0,0,4,15,14,1,0,0,0,3,0,0,0,1,4,1,122,0,0,1,4,1,116,1,0,0,2212,21
%N A356304 The least k >= 0 such that A003415(n) and A276086(n+k) are relatively prime, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.
%H A356304 Antti Karttunen, <a href="/A356304/b356304.txt">Table of n, a(n) for n = 2..11550</a>
%H A356304 Antti Karttunen, <a href="/A356304/a356304.txt">Data supplement: n, a(n) computed for n = 2..65537</a>
%H A356304 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%o A356304 (PARI)
%o A356304 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A356304 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A356304 A356304(n) = { my(k=0,x=A003415(n)); while(gcd(A276086(n+k),x)!=1,k++); (k); };
%Y A356304 Cf. A003415, A276086, A356311 (after its initial zero gives the positions of zeros in this sequence).
%Y A356304 Cf. also A356302, A356305.
%K A356304 nonn
%O A356304 2,5
%A A356304 _Antti Karttunen_, Nov 03 2022