A227068 -id:A227068 - cp's OEIS frontend

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

Henry Bottomley, Jul 12 2000

nonn

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

			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)

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).

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

Cf. A227068 (records).

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

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

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 *)

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

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

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

Edited by Michael Somos, Mar 18 2006

A038549 Least number with exactly n divisors that are at most its square root.

Original entry on oeis.org

1, 4, 12, 24, 36, 60, 192, 120, 180, 240, 576, 360, 1296, 900, 720, 840, 9216, 1260, 786432, 1680, 2880, 15360, 3600, 2520, 6480, 61440, 6300, 6720, 2359296, 5040, 3221225472, 7560, 46080, 983040, 25920, 10080, 206158430208, 32400, 184320

Offset: 1

Tom Verhoeff

nonn

Least number of identical objects that can be arranged in exactly n ways in a rectangle, modulo rotation.
Smallest number which has n distinct unordered factorizations of the form x*y. - Lekraj Beedassy, Jan 09 2008
Note that an upper bound on a(n) is 3*2^(n-1), which is attained at n = 4 and the odd primes in A005382 (primes p such that 2p-1 is also prime). - T. D. Noe, Jul 13 2013

Paul Tek, Table of n, a(n) for n = 1..1000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A038548 (records), A072671, A004778, A086921.

Cf. A227068 (similar, but with limit < sqrt).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a038549 = (+ 1) . fromJust . (`elemIndex` a038548_list)
-- Reinhard Zumkeller, Dec 26 2012

nn = 18; t = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++; c = Length[Select[Divisors[n], # <= Sqrt[n] &]]; If[c > 0 && c <= nn && t[[c]] == 0, t[[c]] = n; found++]]; t (* T. D. Noe, Jul 10 2013 *)

a(n) = min(A005179(2n-1), A005179(2n)).

More terms from David W. Wilson.

cp's OEIS Frontend

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

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