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A332770 a(n) is the number of ways to write A180045(n) as (x*y+1)*(x*z+1) with x > y > z > 1.

%I A332770 #10 Jun 10 2021 21:03:29
%S A332770 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,
%T A332770 2,1,1,1,1,1,1,1,1,2,1,1,3,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,
%U A332770 1,2,1,2,1,1,1,1,2,2,1,1,1,1,1,2,1,1,2
%N A332770 a(n) is the number of ways to write A180045(n) as (x*y+1)*(x*z+1) with x > y > z > 1.
%H A332770 Robert Israel, <a href="/A332770/b332770.txt">Table of n, a(n) for n = 1..10000</a>
%H A332770 Rémy Sigrist, <a href="/A332770/a332770_2.txt">C program for A332770</a>
%e A332770 The first terms, alongside A180045(n), are:
%e A332770   n   a(n)  A180045(n)
%e A332770   --  ----  ---------------------------------------
%e A332770    1     1   28 = (3*2+1)*(3*1+1)
%e A332770    2     1   45 = (4*2+1)*(4*1+1)
%e A332770    3     1   65 = (4*3+1)*(4*1+1)
%e A332770    4     1   66 = (5*2+1)*(5*1+1)
%e A332770    5     1   91 = (6*2+1)*(6*1+1)
%e A332770    6     1   96 = (5*3+1)*(5*1+1)
%e A332770    7     1  117 = (4*3+1)*(4*2+1)
%e A332770    8     1  120 = (7*2+1)*(7*1+1)
%e A332770    9     1  126 = (5*4+1)*(5*1+1)
%e A332770   10     1  133 = (6*3+1)*(6*1+1)
%e A332770   11     1  153 = (8*2+1)*(8*1+1)
%e A332770   12     1  175 = (6*4+1)*(6*1+1)
%e A332770   13     2  176 = (5*3+1)*(5*2+1) = (7*3+1)*(7*1+1)
%p A332770 N:= 20000: # for a(n) where A180045(n) <= N
%p A332770 V:= Vector(N):
%p A332770 for x from 3 while (2*x+1)*(x+1) <= N do
%p A332770   for y from 2 to x-1 while (x*y+1)*(x+1) <= N do
%p A332770     for z from 1 to y-1 do
%p A332770       v:= (x*y+1)*(x*z+1);
%p A332770       if v > N then break fi;
%p A332770       V[v]:= V[v]+1;
%p A332770 od od od:
%p A332770 subs(0=NULL,convert(V,list)); # _Robert Israel_, Jun 10 2021
%o A332770 (C) See Links section.
%Y A332770 Cf. A180045.
%K A332770 nonn
%O A332770 1,13
%A A332770 _Rémy Sigrist_, Feb 23 2020