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

%I A056924 #71 Nov 12 2023 13:27:53
%S A056924 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,
%T A056924 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,
%U A056924 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
%N A056924 Number of divisors of n that are smaller than sqrt(n).
%C A056924 Number of powers of n in product of factors of n if n>1.
%C A056924 Also, the number of solutions to the Pell equation x^2 - y^2 = 4n. - _Ralf Stephan_, Sep 20 2013
%C A056924 If n is a prime or the square of a prime, then a(n)=1.
%C A056924 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
%C A056924 Number of pairs of distinct divisors (d,n/d) of n, with d<n/d. - _Wesley Ivan Hurt_, Nov 09 2023
%H A056924 T. D. Noe, <a href="/A056924/b056924.txt">Table of n, a(n) for n=1..10000</a>
%H A056924 Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.1016/j.jnt.2016.06.007">New convolutions for the number of divisors</a>, Journal of Number Theory, Vol. 170 (2016), pp. 17-34.
%H A056924 S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph invariants based on the divides relation and ordered by prime signatures</a>, arXiv:1405.5283 [math.NT], 2014, eq. (2.29).
%F A056924 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.
%F A056924 a(n) = A000005(n) - A038548(n). - _Labos Elemer_, Apr 19 2002
%F A056924 G.f.: Sum_{k>0} x^(k^2+k)/(1-x^k). - _Michael Somos_, Mar 18 2006
%F A056924 a(n) = (1/2) * Sum_{d|n} (1 - [d = n/d]), where [ ] is the Iverson bracket. - _Wesley Ivan Hurt_, Jan 28 2021
%e A056924 a(16)=2 since the divisors of 16 are 1,2,4,8,16 of which 2 are less than sqrt(16) = 4.
%e A056924 From _Labos Elemer_, Apr 19 2002: (Start)
%e A056924 n=96: a(96) = Card[{1,2,3,4,6,8}] = 6 = Card[{12,16,24,32,48,96}];
%e A056924 n=225: a(225) = Card[{1,3,5,9}] = Card[{15,25,45,7,225}]-1. (End)
%p A056924 with(numtheory); A056924 := n->floor(tau(n)/2); seq(A056924(k),k=1..100); # _Wesley Ivan Hurt_, Jun 14 2013
%t A056924 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 *)
%t A056924 Table[Length[Select[Divisors[n], # < Sqrt[n] &]], {n, 100}] (* _T. D. Noe_, Jul 11 2013 *)
%t A056924 a[n_] := Floor[DivisorSigma[0, n]/2]; Array[a, 100] (* _Amiram Eldar_, Jun 26 2022 *)
%o A056924 (PARI) a(n)=if(n<1, 0, numdiv(n)\2) /* _Michael Somos_, Mar 18 2006 */
%o A056924 (Haskell)
%o A056924 a056924 = (`div` 2) . a000005  -- _Reinhard Zumkeller_, Jul 12 2013
%o A056924 (Python)
%o A056924 from sympy import divisor_count
%o A056924 def A056924(n): return divisor_count(n)//2 # _Chai Wah Wu_, Jun 25 2022
%Y A056924 Cf. A038548, A000203, A000005, A070038, A070039.
%Y A056924 Cf. A227068 (records).
%K A056924 nonn
%O A056924 1,6
%A A056924 _Henry Bottomley_, Jul 12 2000
%E A056924 Edited by _Michael Somos_, Mar 18 2006