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

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, 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, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2

Offset: 1

Author

Rémy Sigrist, Feb 23 2020

Keywords

• nonn

Examples

			The first terms, alongside A180045(n), are:
  n   a(n)  A180045(n)
  --  ----  ---------------------------------------
   1     1   28 = (3*2+1)*(3*1+1)
   2     1   45 = (4*2+1)*(4*1+1)
   3     1   65 = (4*3+1)*(4*1+1)
   4     1   66 = (5*2+1)*(5*1+1)
   5     1   91 = (6*2+1)*(6*1+1)
   6     1   96 = (5*3+1)*(5*1+1)
   7     1  117 = (4*3+1)*(4*2+1)
   8     1  120 = (7*2+1)*(7*1+1)
   9     1  126 = (5*4+1)*(5*1+1)
  10     1  133 = (6*3+1)*(6*1+1)
  11     1  153 = (8*2+1)*(8*1+1)
  12     1  175 = (6*4+1)*(6*1+1)
  13     2  176 = (5*3+1)*(5*2+1) = (7*3+1)*(7*1+1)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, C program for A332770

Crossrefs

Cf. A180045.

Programs

C

See Links section.

Maple

N:= 20000: # for a(n) where A180045(n) <= N
V:= Vector(N):
for x from 3 while (2*x+1)*(x+1) <= N do
  for y from 2 to x-1 while (x*y+1)*(x+1) <= N do
    for z from 1 to y-1 do
      v:= (x*y+1)*(x*z+1);
      if v > N then break fi;
      V[v]:= V[v]+1;
od od od:
subs(0=NULL,convert(V,list)); # Robert Israel, Jun 10 2021