%I #6 Jul 10 2016 17:12:17
%S 1,6,18,12,24,60,156,84,144,120,816,336,360,1224,840,924,2184,1800,
%T 2640,7200,1260,3960,7140,8400,3780,5040,2520,9360,12600,20160,11340,
%U 10080,15120,19656,16380,41580,18480,48720,34320,25200,54600,27720,87360,134640,60060,73920,32760,43680,159600,143640,55440,85800,96096,65520,131040,120120,157080,154440,98280,191520,166320
%N Least number, m, such that m^2 is expressible in just n ways as (p+1)(q+1) where p and q are distinct primes.
%C Records: 1, 6, 12, 24, 60, 84, 120, 336, 360, 840, ..., .
%e a(0) = 1 since 1 is not expressible as (p+1)(q+1), in fact no odd number is expressible this way;
%e a(1) = 6 since 6^2 = 36 = (2+1)(11+1);
%e a(2) = 18 since 18^2 = 324 = (2+1)(107+1) = (5+1)(53+1);
%e a(3) = 12 since 12^2 = 144 = (2+1)(47+1) = (5+1)(23+1) = (7+1)(17+1); etc.
%t (* first compute A274876: f[n_] := f[n] = Block[{c = 0, p = 2}, While[p < 2n -1, If[ PrimeQ[(2n)^2/(p +1) -1], c++]; p = NextPrime@ p]; c]; f[0] = 1; then *) Table[(2Select[ Range@ 5000, f@# == n &])[[1]], {n, 0, 10}]
%Y Cf. A274848, A274877.
%K nonn
%O 0,2
%A _Zak Seidov_ and _Robert G. Wilson v_, Jul 10 2016