A056924 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

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 d<n/d. - Wesley Ivan Hurt, Nov 09 2023

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).

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

EXAMPLE

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)

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 *)

PROG

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

(Haskell)

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

(Python)

from sympy import divisor_count

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

CROSSREFS

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

Cf. A227068 (records).