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A345215 a(n) is the least positive number k that can be written in exactly k ways as (x*y+1)*(x*z+1) with x > y > z > 1.

%I A345215 #4 Jun 10 2021 22:33:52
%S A345215 1,28,176,561,2701,7381,29161,51681,115921,390241,260281,924001,
%T A345215 1334161,1413721,1038961,3178981,8826301,3000025,16597441,33882241,
%U A345215 12708361,22589281,31375081,63095761,90336961
%N A345215 a(n) is the least positive number k that can be written in exactly k ways as (x*y+1)*(x*z+1) with x > y > z > 1.
%C A345215 For n > 1, a(n) = A180045(k) where A332770(k) = n is the first appearance of n in A332770.
%e A345215 a(3) = 561 since 176 = (8*4+1)*(8*2+1) = (10*5+1)*(10*1+1) = (16*2+1)*(16*1+1) is the first number that can be obtained in exactly 3 ways.
%p A345215 N:= 10^8:
%p A345215 V:= Vector(N,datatype=integer[4]):
%p A345215 for x from 3 while (2*x+1)*(x+1) <= N do
%p A345215   for y from 2 to x-1 while (x*y+1)*(x+1) <= N do
%p A345215     for z from 1 to y-1 do
%p A345215       v:= (x*y+1)*(x*z+1);
%p A345215       if v > N then break fi;
%p A345215       V[v]:= V[v]+1;
%p A345215 od od od:
%p A345215 R:= Array(0..26):
%p A345215 for j from 1 to N do
%p A345215   v:= V[j]; if R[v] = 0 then R[v]:= j fi
%p A345215 od:
%p A345215 convert(R[0..24],list);
%Y A345215 Cf. A180045, A332770.
%K A345215 nonn
%O A345215 0,2
%A A345215 _Robert Israel_, Jun 10 2021