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a(n) = gcd(A011772(n), A344875(n)).
8

%I #14 Jun 20 2021 19:19:08

%S 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 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 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 a(n) = gcd(A011772(n), A344875(n)).

%H Antti Karttunen, <a href="/A344969/b344969.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = gcd(A011772(n), A344875(n)).

%F a(n) = gcd(A011772(n), A344876(n)) = gcd(A344875(n), A344876(n)) = gcd(A011772(n), A344973(n)).

%t A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];

%t A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];

%t a[n_] := GCD[A011772[n], A344875[n]];

%t Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)

%o (PARI)

%o 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 A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };

%o A344969(n) = gcd(A011772(n), A344875(n));

%Y Cf. A011772, A344875, A344876, A344970, A344971, A344972, A344973.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 03 2021