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A348039 a(n) = gcd(A003557(n), A327564(n)).

%I A348039 #15 Oct 29 2021 12:19:38
%S A348039 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A348039 1,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A348039 1,1,1,12,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,6
%N A348039 a(n) = gcd(A003557(n), A327564(n)).
%C A348039 There is interesting regularity in the scatter plot.
%H A348039 Antti Karttunen, <a href="/A348039/b348039.txt">Table of n, a(n) for n = 1..65537</a>
%F A348039 a(n) = gcd(A003557(n), A327564(n)).
%F A348039 a(n) = A348036(n) / A007947(n).
%F A348039 a(n) = A003557(n) / A348037(n).
%F A348039 a(n) = A327564(n) / A348038(n).
%t A348039 {1}~Join~Array[GCD @@ Map[Times @@ # &, Transpose@ Map[{#1^(#2 - 1), (#1 + 1)^(#2 - 1)} & @@ # &, FactorInteger[#]]] &, 105, 2] (* _Michael De Vlieger_, Oct 20 2021 *)
%o A348039 (PARI)
%o A348039 A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
%o A348039 A007947(n) = factorback(factorint(n)[, 1]);
%o A348039 A348036(n) = gcd(n, A003968(n));
%o A348039 A348039(n) = (A348036(n)/A007947(n));
%o A348039 (PARI)
%o A348039 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A348039 A327564(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1]++; f[k, 2]--); factorback(f); }; \\ From A327564
%o A348039 A348039(n) = gcd(A003557(n), A327564(n));
%Y A348039 Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038.
%Y A348039 Cf. A347960 (positions of terms > 1).
%K A348039 nonn,look
%O A348039 1,36
%A A348039 _Antti Karttunen_, Oct 19 2021