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A344973 a(n) = A344875(n) mod A011772(n).

%I A344973 #15 Jun 20 2021 19:19:22
%S A344973 0,0,0,0,0,0,0,0,0,0,0,6,0,4,3,0,0,0,0,13,0,8,0,0,0,0,0,0,0,9,0,0,9,0,
%T A344973 10,0,0,16,0,0,0,16,0,6,5,20,0,30,0,0,15,6,0,24,0,42,0,0,0,11,0,28,21,
%U A344973 0,23,5,0,0,21,12,0,57,0,0,0,14,18,0,0,60,0,0,0,36,30,40,27,22,0,26,7,16,0,44,15,0,0,0,36
%N A344973 a(n) = A344875(n) mod A011772(n).
%H A344973 Antti Karttunen, <a href="/A344973/b344973.txt">Table of n, a(n) for n = 1..11264</a>
%H A344973 Antti Karttunen, <a href="/A344973/a344973.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A344973 a(n) = A344875(n) mod A011772(n) = A344876(n) mod A011772(n).
%t A344973 b[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe;
%t A344973      If[p == 2, 2^(1 + e) - 1, p^e - 1], {pe, FactorInteger[n]}]]];
%t A344973 c[n_] := Module[{m = 1}, While[Not[IntegerQ[m (m + 1)/(2 n)]], m++]; m];
%t A344973 a[n_] := Mod[b[n], c[n]];
%t A344973 Array[a, 100] (* _Jean-François Alcover_, Jun 12 2021 *)
%o A344973 (PARI)
%o A344973 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 A344973 A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
%o A344973 A344973(n) = (A344875(n)%A011772(n));
%Y A344973 Cf. A011772, A344875, A344876, A344969, A344970, A344971, A344972.
%Y A344973 Cf. A344974 (positions of zeros).
%K A344973 nonn
%O A344973 1,12
%A A344973 _Antti Karttunen_, Jun 04 2021