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 A056924 Number of divisors of n that are smaller than sqrt(n). 72
 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 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, 2016, vol. 170, 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), (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 *) 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 CROSSREFS Cf. A038548, A000203, A000005, A070038, A070039. Cf. A227068 (records). Sequence in context: A341596 A099042 A140774 * A342083 A316364 A318357 Adjacent sequences:  A056921 A056922 A056923 * A056925 A056926 A056927 KEYWORD nonn AUTHOR Henry Bottomley, Jul 12 2000 EXTENSIONS Edited by Michael Somos, Mar 18 2006 STATUS approved

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Last modified May 11 03:49 EDT 2021. Contains 343784 sequences. (Running on oeis4.)