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%I #40 Oct 21 2023 05:14:01
%S 1,-1,3,-3,1,-5,7,-3,9,-9,3,-11,9,-1,15,-15,5,-9,19,-7,21,-21,3,-23,9,
%T -9,27,-15,9,-29,31,-3,21,-33,11,-35,37,-9,15,-39,1,-41,27,-15,45,-21,
%U 15,-27,49,-5,51,-51,9,-53,55,-19,57,-33,7,-27
%N Let 2*n+1 have prime factorization Prod_i p_i^k_i; then a(n) = Prod_i v_i^k_i, where v_i = (1+p_i)/2 if p_i == 1 (mod 4), v_i = (1-p_i)/2 if p_i == 3(mod 4).
%C Completely multiplicative.
%C If n = prime(k), a(n) = A271974(k).
%H Dimitris Valianatos, <a href="/A272295/a272295.txt">Comments on this sequence</a>, April 25 2016
%F Sum_{n >= 1, n not divisible by 2 or 3} 1/a(n) = 1.
%F Conjecture: Sum_{n >= 1, n not divisible by 2 or 3} (mb(n)/a(n))^2 = 7/5 = 1.4;
%F mb(n) is the moebius function.
%e For n=35=5*7, 5-> 3 and 7->-3 so v(35)=3*(-3)=-9, for n=77=7*11, 7->(-3) and 11->(-5) so v(77)=(-3)*(-5)=35.
%t Table[Times @@ Apply[Power[#1, #2] &, Transpose@ MapAt[Which[Mod[#, 4] == 1, (1 + #)/2, Mod[#, 4] == 3, (1 - #)/2, True, 0] & /@ # &, Transpose@ FactorInteger@ n, 1], 1], {n, 1, 120, 2}] (* _Michael De Vlieger_, Apr 24 2016 *)
%o (PARI) a(n)=if(n%2 == 0, return(0));fa=factorint(n);dv=fa[,1];pl=#dv;ml=fa[,2];g=1;for(i=1,pl,ds=dv[i];v=1;if(ds%4==1,v*=(1+ds)\2,v*=(1-ds)\2);for(k=1,ml[i],g*=v));return(g);
%o (PARI) {
%o forstep(n=1,120,2,
%o fa=factorint(n);dv=fa[,1];pl=#dv;ml=fa[,2];
%o g=1;
%o for(i=1,pl,
%o ds=dv[i];v=1;
%o if(ds%4==1,v*=(1+ds)\2,v*=(1-ds)\2);
%o for(k=1,ml[i],g*=v)
%o );
%o print1(g", ")
%o );
%o }
%o (PARI) a(n) = {my(f = factor(2*n-1)); for (k=1, #f~, if (f[k,1] == 2, f[k,1] = 0, if (f[k,1] % 4 == 1, f[k,1] = (1+f[k,1])/2, f[k,1] = (1-f[k,1])/2));); factorback(f);} \\ _Michel Marcus_, May 02 2016
%Y Cf. A271974.
%K sign,mult
%O 1,3
%A _Dimitris Valianatos_, Apr 24 2016
%E Edited by _Franklin T. Adams-Watters_, Apr 24 2016 and by _N. J. A. Sloane_, May 27 2016
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