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A344970 a(n) = A011772(n) / gcd(A011772(n), A344875(n)).

%I A344970 #18 Jun 30 2021 19:48:40
%S A344970 1,1,1,1,1,1,1,1,1,1,1,4,1,7,5,1,1,1,1,15,1,11,1,1,1,1,1,1,1,5,1,1,11,
%T A344970 1,7,1,1,19,1,1,1,5,1,16,9,23,1,16,1,1,17,13,1,9,1,8,1,1,1,15,1,31,9,
%U A344970 1,25,11,1,1,23,5,1,21,1,1,1,4,7,1,1,16,1,1,1,4,17,43,29,16,1,35,13,23,1,47,19,1,1,1,11
%N A344970 a(n) = A011772(n) / gcd(A011772(n), A344875(n)).
%C A344970 Denominator of the ratio A344875(n)/A011772(n): 1/1, 3/3, 2/2, 7/7, 4/4, 6/3, 6/6, 15/15, 8/8, 12/4, 10/10, 14/8, 12/12, 18/7, 8/5, 31/31, 16/16, 24/8, 18/18, 28/15, 12/6, 30/11, ..., = 1/1, 1/1, 1/1, 1/1, 1/1, 2/1, 1/1, 1/1, 1/1, 3/1, 1/1, 7/4, 1/1, 18/7, 8/5, 1/1, 1/1, 3/1, 1/1, 28/15, 2/1, 30/11, etc.
%H A344970 Antti Karttunen, <a href="/A344970/b344970.txt">Table of n, a(n) for n = 1..16384</a>
%H A344970 Antti Karttunen, <a href="/A344970/a344970.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A344970 a(n) = A011772(n) / A344969(n) = A011772(n) / gcd(A011772(n), A344875(n)).
%t A344970 A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
%t A344970 A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
%t A344970 a[n_] := A011772[n]/GCD[A011772[n], A344875[n]];
%t A344970 Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)
%o A344970 (PARI)
%o A344970 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 A344970 A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
%o A344970 A344970(n) = { my(u=A011772(n)); (u/gcd(u, A344875(n))); };
%Y A344970 Cf. A011772, A344875, A344969, A344971 (numerators), A344972 (ratio floored down), A344974 (positions of ones), A344980 (of terms > 1).
%K A344970 nonn,frac
%O A344970 1,12
%A A344970 _Antti Karttunen_, Jun 04 2021