A056924 Number of divisors of n that are smaller than sqrt(n).

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

Offset: 1

Author

Henry Bottomley, Jul 12 2000

Keywords

• nonn

Comments

Number of powers of n in product of factors of n if n>1.

Also, the number of solutions to the Pell equation x^2 - y^2 = 4n. - Ralf Stephan, Sep 20 2013

If n is a prime or the square of a prime, then a(n)=1.

Number of positive integer solutions to the equation x^2 + k*x - n = 0, for all k in 1 <= k <= n. - Wesley Ivan Hurt, Dec 27 2020

Number of pairs of distinct divisors (d,n/d) of n, with dWesley Ivan Hurt, Nov 09 2023

Examples

			a(16)=2 since the divisors of 16 are 1,2,4,8,16 of which 2 are less than sqrt(16) = 4.
From _Labos Elemer_, Apr 19 2002: (Start)
n=96: a(96) = Card[{1,2,3,4,6,8}] = 6 = Card[{12,16,24,32,48,96}];
n=225: a(225) = Card[{1,3,5,9}] = Card[{15,25,45,7,225}]-1. (End)
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, Vol. 170 (2016), pp. 17-34.
• S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT], 2014, eq. (2.29).

Crossrefs

Cf. A038548, A000203, A000005, A070038, A070039.

Cf. A227068 (records).

Programs

Haskell

a056924 = (`div` 2) . a000005  -- Reinhard Zumkeller, Jul 12 2013

Maple

with(numtheory); A056924 := n->floor(tau(n)/2); seq(A056924(k),k=1..100); # Wesley Ivan Hurt, Jun 14 2013

Mathematica

di[x_] := Divisors[x] lds[x_] := Ceiling[DivisorSigma[0, x]/2] rd[x_] := Reverse[Divisors[x]] td[x_] := Table[Part[rd[x], w], {w, 1, lds[x]}] sud[x_] := Apply[Plus, td[x]] Table[DivisorSigma[0, w]-lds[w], {w, 1, 128}] (* Labos Elemer, Apr 19 2002 *)
Table[Length[Select[Divisors[n], # < Sqrt[n] &]], {n, 100}] (* T. D. Noe, Jul 11 2013 *)
a[n_] := Floor[DivisorSigma[0, n]/2]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

PARI

a(n)=if(n<1, 0, numdiv(n)\2) /* Michael Somos, Mar 18 2006 */

Python

from sympy import divisor_count
def A056924(n): return divisor_count(n)//2 # Chai Wah Wu, Jun 25 2022

Formula

For n>1, a(n) = floor[log(A007955(n))/log(n)] = log(A056925(n))/log(n) = floor[d(n)/2] = floor[A000005(n)/2] = ( A000005(n)-A010052(n) )/2.

a(n) = A000005(n) - A038548(n). - Labos Elemer, Apr 19 2002

G.f.: Sum_{k>0} x^(k^2+k)/(1-x^k). - Michael Somos, Mar 18 2006

a(n) = (1/2) * Sum_{d|n} (1 - [d = n/d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

Extensions

Edited by Michael Somos, Mar 18 2006