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A344969 a(n) = gcd(A011772(n), A344875(n)).

%I A344969 #14 Jun 20 2021 19:19:08
%S A344969 1,3,2,7,4,3,6,15,8,4,10,2,12,1,1,31,16,8,18,1,6,1,22,15,24,12,26,7,
%T A344969 28,3,30,63,1,16,2,8,36,1,12,15,40,4,42,2,1,1,46,2,48,24,1,3,52,3,10,
%U A344969 6,18,28,58,1,60,1,3,127,1,1,66,16,1,4,70,3,72,36,24,14,3,12,78,4,80,40,82,12,2,1,1,2,88,1,1,1,30
%N A344969 a(n) = gcd(A011772(n), A344875(n)).
%H A344969 Antti Karttunen, <a href="/A344969/b344969.txt">Table of n, a(n) for n = 1..10000</a>
%F A344969 a(n) = gcd(A011772(n), A344875(n)).
%F A344969 a(n) = gcd(A011772(n), A344876(n)) = gcd(A344875(n), A344876(n)) = gcd(A011772(n), A344973(n)).
%t A344969 A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
%t A344969 A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
%t A344969 a[n_] := GCD[A011772[n], A344875[n]];
%t A344969 Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)
%o A344969 (PARI)
%o A344969 A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
%o A344969 A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
%o A344969 A344969(n) = gcd(A011772(n), A344875(n));
%Y A344969 Cf. A011772, A344875, A344876, A344970, A344971, A344972, A344973.
%K A344969 nonn
%O A344969 1,2
%A A344969 _Antti Karttunen_, Jun 03 2021