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A056924
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Number of divisors of n that are smaller than sqrt(n).
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78
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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
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OFFSET
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1,6
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = (1/2) * Sum_{d|n} (1 - [d = n/d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021
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EXAMPLE
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a(16)=2 since the divisors of 16 are 1,2,4,8,16 of which 2 are less than sqrt(16) = 4.
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)
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n<1, 0, numdiv(n)\2) /* Michael Somos, Mar 18 2006 */
(Haskell)
(Python)
from sympy import divisor_count
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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