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A356305 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 A356305 #9 Nov 04 2022 11:26:23
%S A356305 0,1,0,0,0,0,1,0,2,3,0,0,0,0,1,1,0,0,0,0,2,17,0,0,0,21,25,2,0,0,0,0,0,
%T A356305 5,0,5,6,0,13,1,0,0,0,0,2,2,11,0,20,21,19,17,24,0,0,1,0,1,0,0,0,0,1,2,
%U A356305 4,5,0,0,2,1,0,0,0,0,1,10,12,5,0,0,0,3,0,0,0,1,25,1,84,0,0,1,2,1,65,5,0,0,69,8,96
%N A356305 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 A356305 Antti Karttunen, <a href="/A356305/b356305.txt">Table of n, a(n) for n = 0..11550</a>
%H A356305 Antti Karttunen, <a href="/A356305/a356305.txt">Data supplement: n, a(n) computed for n = 0..65537</a>
%H A356305 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%o A356305 (PARI)
%o A356305 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A356305 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A356305 A356305(n) = { my(k=0,x=A003415(n)); while(gcd(A276086(n-k),x)!=1,k++); (k); };
%Y A356305 Cf. A003415, A276086, A356311 (positions of 0's).
%Y A356305 Cf. also A356303, A356304.
%K A356305 nonn,look
%O A356305 0,9
%A A356305 _Antti Karttunen_, Nov 03 2022