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A353573 Greatest common divisor of A342002 and its shifted variant, where A342002(n) = A003415(A276086(n)) / A003557(A276086(n)) and A276086 is the primorial base exp-function.

%I A353573 #9 Apr 29 2022 17:18:41
%S A353573 0,1,1,1,2,1,1,1,4,1,1,1,2,1,1,1,8,1,3,1,2,1,1,1,4,1,1,1,2,1,1,1,2,1,
%T A353573 1,1,6,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,3,1,2,1,1,1,2,1,1,1,4,1,1,1,
%U A353573 2,1,1,1,12,1,1,1,2,1,1,1,8,1,1,1,2,1,1,1,4,1,3,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1
%N A353573 Greatest common divisor of A342002 and its shifted variant, where A342002(n) = A003415(A276086(n)) / A003557(A276086(n)) and A276086 is the primorial base exp-function.
%H A353573 Antti Karttunen, <a href="/A353573/b353573.txt">Table of n, a(n) for n = 0..30030</a>
%H A353573 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A353573 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A353573 a(n) = gcd(A342002(n), A353572(n)).
%F A353573 a(n) = gcd(A353571(A276086(n)), A342001(A276086(n))).
%F A353573 For all n >= 1, a(n) = A342002(n) / A353574(n).
%o A353573 (PARI)
%o A353573 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A353573 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A353573 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A353573 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A353573 A342001(n) = (A003415(n) / A003557(n));
%o A353573 A353571(n) = A342001(A003961(n));
%o A353573 A353573(n) = gcd(A353571(A276086(n)), A342001(A276086(n)));
%Y A353573 Cf. A003415, A003557, A003961, A342001, A342002, A353572, A353574.
%K A353573 nonn
%O A353573 0,5
%A A353573 _Antti Karttunen_, Apr 29 2022