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a(n)=x is the least integer such that gcd(sigma(x), sigma(x+1)) = 2*n.
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%I #32 Jan 28 2023 12:37:11

%S 13,6,5,27,57,22,12,93,89,236,1032,14,467,83,58,308,201,118,147,56,82,

%T 1204,6301,69,596,1142,106,91,4167,87,432,381,393,1407,348,70,5912,

%U 453,233,417,13692,166,56493,1118,88,6987,54048,154,1843,4490,6833,2574,633,689,1538

%N a(n)=x is the least integer such that gcd(sigma(x), sigma(x+1)) = 2*n.

%C Bisection of A084307.

%H Michel Marcus, <a href="/A322569/b322569.txt">Table of n, a(n) for n = 1..1000</a>

%t Module[{nn=60000,g},g=GCD@@@Partition[DivisorSigma[1,Range[nn]],2,1];Table[ Position[ g,2n,1,1],{n,55}]]//Flatten (* _Harvey P. Dale_, Jan 28 2023 *)

%o (PARI) a(n) = my(x=1); while(gcd(sigma(x), sigma(x+1)) != 2*n, x++); x;

%o (Magma) sol:=[]; for n in [1..55] do k:=1; while Gcd(DivisorSigma(1,k),DivisorSigma(1,k+1)) ne 2*n do k:=k+1; end while; Append(~sol,k); end for; sol; // _Marius A. Burtea_, Aug 29 2019

%Y Cf. A000203, A060780, A084307.

%K nonn

%O 1,1

%A _Michel Marcus_, Aug 29 2019