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Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.
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%I #29 Jul 01 2023 11:00:08

%S 3,6,18,40,127251,198,26890623,918,17298,6640,25269208984375,3400,

%T 3900566650390623,640062,8418573,18088,1164385682220458984373,41650,

%U 69528379848480224609373,128464,34084859373,12164094,150509919493198394775390625,90270,418514293125,64505245696

%N Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.

%e For n=1, numdiv(3) = numdiv(5) = 2 = 2*1, and no number < 3 satisfies this, hence a(1) = 3.

%t a={}; n=1; nmax=10; For[k=1, n<=nmax, k++, If[DivisorSigma[0, k] == DivisorSigma[0, k+2] == 2n, AppendTo[a, k]; k=1; n++]]; a (* _Stefano Spezia_, Aug 26 2022 *)

%t Flatten[Table[SequencePosition[DivisorSigma[0,Range[27*10^6]],{2n,_,2n},1],{n,10}],1][[;;,1]] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant but the program will take a long time to run. *) (* _Harvey P. Dale_, Jul 01 2023 *)

%o (PARI) a(n)=for(k=1,+oo,if(numdiv(k)==2*n&&numdiv(k+2)==2*n,return(k)))

%Y Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), A356743 (m=6), A356744 (m=8).

%Y Cf. A000005 (d(n)), A003680, A005238, A006558, A006601, A062832, A067888, A067889, A075036.

%K nonn

%O 1,1

%A _Jean-Marc Rebert_, Aug 26 2022

%E More terms from _Jinyuan Wang_, Aug 28 2022