A095686 Half the number of divisors of nonsquares (A000037).

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

Offset: 1

Author

Lekraj Beedassy, Jul 05 2004

Keywords

• nonn

Comments

The first occurrence of n in the sequence corresponds to the nonsquare = A003680(n) = A005179(2n).

Also number of unordered divisor pairs (d,n/d) for n=A000037. - Lekraj Beedassy, Jul 30 2004

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Programs

Mathematica

m=11; DivisorSigma[0, Complement[Range[m^2], Range[m]^2]]/2 (* Amiram Eldar, Nov 05 2019 *)

PARI

lista(nn) = {for (i = 1, nn, if (! issquare(i), print1(numdiv(i)/2, ", ")););} \\ Michel Marcus, Jul 27 2013

Python

from math import isqrt
from sympy import divisor_count
def A095686(n): return divisor_count(n+(k:=isqrt(n))+int(n>k*(k+1)))>>1 # Chai Wah Wu, Jun 05 2025

Formula

a(n)=A000005(A000037(n))/2.

Extensions

Corrected and extended by Ray Chandler, Jul 09 2004