A033831 Number of numbers d dividing n such that d >= 3 and n/d <= d-2.

0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 6, 1, 2, 3, 3, 2, 4, 1, 3, 2, 4, 1, 5, 1, 2, 3, 3, 2, 4, 1, 5, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 4, 1, 4, 4

Offset: 1

Author

Naohiro Nomoto

Keywords

• easy
• nonn

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002378.

Programs

Maple

with(numtheory): for n from 1 to 200 do it := divisors(n): count := 0: for i from 1 to nops(it) do if it[i]>=3 and 1<=n/it[i] and n/it[i]<=(it[i]-2) then count := count+1 fi :od: printf(`%d,`,count) od:

Mathematica

a[n_] := DivisorSum[n, 1&, # > 2 && n/# < #-1 &]; Array[a, 100] (* Amiram Eldar, Jun 11 2019 *)

PARI

a(n) = sumdiv(n, d, (d>=3) && (q=n/d) && (q>=1) && (q<=d-2)); \\ Michel Marcus, Nov 05 2014

PARI

a033831(n) = numdiv(n)\2 - issquare(4*n+1); \\ Max Alekseyev, Oct 09 2023

Formula

G.f.: Sum_{n>=1} x^(n^2+2*n)/(1-x^n). - Vladeta Jovovic, May 13 2008

a(n) = floor(A000005(n)/2) - 1 if n is oblong (A002378); and floor(A000005(n)/2) otherwise. - Max Alekseyev, Oct 09 2023

Extensions

More terms from James Sellers, Jun 20 2000