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A373367 a(n) is the greatest common divisor of A001414(n), A003415(n), and A276085(n).

%I A373367 #13 Jun 04 2024 10:52:48
%S A373367 0,1,1,2,1,1,1,3,2,7,1,1,1,1,8,4,1,1,1,1,2,1,1,1,2,1,3,1,1,1,1,5,2,1,
%T A373367 12,2,1,1,8,1,1,1,1,1,1,1,1,1,2,1,4,1,1,1,8,1,2,1,1,2,1,1,1,6,6,1,1,1,
%U A373367 2,1,1,1,1,1,1,1,6,1,1,1,4,1,1,2,2,1,8,1,1,1,20,1,2,1,12,1,1,1,1,14,1,1,1,1,1
%N A373367 a(n) is the greatest common divisor of A001414(n), A003415(n), and A276085(n).
%C A373367 All sequences that give the positions of multiples of some natural number k in this sequence are closed under multiplication because the constituent sequences A001414, A003415, and A276085 also have the same property.
%C A373367 A345452 gives the positions of even terms in this sequence, because it gives them for A373362, and because for A373145 and A373364 the positions of even terms are given by A368998 (union of A345452 and 2*A358776) and A373375 (union of A345452 and 8*A345452), thus both are supersets of A345452.
%H A373367 Antti Karttunen, <a href="/A373367/b373367.txt">Table of n, a(n) for n = 1..100000</a>
%F A373367 a(n) = gcd(A373145(n), A373362(n)) = gcd(A373145(n), A373364(n)) = gcd(A373362(n), A373364(n)).
%o A373367 (PARI)
%o A373367 A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
%o A373367 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A373367 A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
%o A373367 A373367(n) = gcd([A001414(n), A003415(n), A276085(n)]);
%Y A373367 Cf. A001414, A003415, A276085, A345452 (gives the positions of even terms).
%Y A373367 Greatest common divisor of any two of these three: A373145, A373362, A373364.
%K A373367 nonn
%O A373367 1,4
%A A373367 _Antti Karttunen_, Jun 03 2024