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 A038548 Number of divisors of n that are at most sqrt(n). 191
 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 5, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 4, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 3, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 4, 1, 4, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of ways to arrange n identical objects in a rectangle, modulo rotation. Number of unordered solutions of x*y = n. - Colin Mallows, Jan 26 2002 Number of ways to write n-1 as n-1 = x*y + x + y, 0<=x<=y<=n. - Benoit Cloitre, Jun 23 2002 Also number of values for x where x+2n and x-2n are both squares (e.g., if n=9, then 18+18 and 18-18 are both squares, as are 82+18 and 82-18 so a(9)=2); this is because a(n) is the number of solutions to n=k(k+r) in which case if x=r^2+2n then x+2n=(r+2k)^2 and x-2n=r^2 (cf. A061408). - Henry Bottomley, May 03 2001 Also number of sums of sequences of consecutive odd numbers or consecutive even numbers including sequences of length 1 (e.g., 12 = 5+7 or 2+4+6 or 12 so a(12)=3). - Naohiro Nomoto, Feb 26 2002 Number of partitions whose consecutive parts differ by exactly two. a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 06 2005 Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly twice. Example: a(12)=3 because we have [3,3,2,2,1,1],[2,2,2,2,2,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006 a(n) is also the number of nonnegative integer solutions of the Diophantine equation 4 x^2-y^2=16 n. For example, a(24)=4 because there are 4 solutions: (x,y)=(10,4),(11,10),(14,20),(25,46). - N-E. Fahssi, Feb 27 2008 a(n) is the number of even divisors of 2*n that are <=sqrt(2*n). - Joerg Arndt, Mar 04 2010 First differences of A094820. - John W. Layman, Feb 21 2012 a(n) = #{k: A027750(n,k) <= A000196(n)}; a(A008578(n)) = 1; a(A002808(n)) > 1. - Reinhard Zumkeller, Dec 26 2012 Row lengths of the tables in A161906 and A161908. - Reinhard Zumkeller, Mar 08 2013 Number of positive integers in the sequence defined by x_0 = n, x_(k+1) = (k+1)*(x_k-2)/(k+2) or equivalently by x_k = n/(k+1) - k. - Luc Rousseau, Mar 03 2018 Expanding the first comment: Number of rectangles with area n and integer side lengths, modulo rotation. Also number of 2D grids of n congruent squares, in a rectangle, modulo rotation (cf. A000005 for rectangles instead of squares; cf. A034836 for the 3D case). - Manfred Boergens, Jun 08 2021 Number of divisors of n that have an even number of prime divisors (counted with multiplicity), or in other words, number of terms of A028260 that divide n. - Antti Karttunen, Apr 17 2022 REFERENCES George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, page 18, exer. 21, 22. 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. Christopher Briggs, Y. Hirano, and H. Tsutsui, Positive Solutions to Some Systems of Diophantine Equations, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.4. 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, (2.27). Madeline Locus Dawsey, Matthew Just and Robert Schneider, A "supernormal" partition statistic, arXiv:2107.14284 [math.NT], 2021. See Table 2 p. 21. T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6. FORMULA a(n) = ceiling(d(n)/2), where d(n) = number of divisors of n (A000005). a(2k) = A034178(2k)+A001227(k). a(2k+1) = A034178(2k+1). - Naohiro Nomoto, Feb 26 2002 G.f.: sum(k>=1, x^(k^2)/(1-x^k)). - Jon Perry, Sep 10 2004 Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/2. - Christian G. Bower, Jun 06 2005 [corrected by Vaclav Kotesovec, Aug 19 2019] a(n) = (A000005(n) + A010052(n))/2. - Omar E. Pol, Jun 23 2009 a(n) = A034178(4*n). - Michael Somos, May 11 2011 2*a(n) = A161841(n). - R. J. Mathar, Mar 07 2021 a(n) = A000005(n) - A056924(n) = A056924(n) + A010052(n) = Sum_{d|n} A065043(d). - Antti Karttunen, Apr 17 2022 Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022 EXAMPLE a(4) = 2 since 4 = 2 * 2 = 4 * 1. Also A034178(4*4) = 2 since 16 = 4^2 - 0^2 = 5^2 - 3^2. - Michael Somos, May 11 2011 x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + ... MAPLE with(numtheory): A038548 := n->ceil(sigma(n)/2); MATHEMATICA Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] (* Robert G. Wilson v, Mar 02 2009 *) Table[Count[Divisors[n], _?(#<=Sqrt[n]&)], {n, 110}] (* Harvey P. Dale, Jul 10 2021 *) PROG (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*d <= n))} /* Michael Somos, Jan 25 2005 */ (PARI) a(n)=ceil(numdiv(n)/2) \\ Charles R Greathouse IV, Sep 28 2012 (Haskell) a038548 n = length \$ takeWhile (<= a000196 n) \$ a027750_row n -- Reinhard Zumkeller, Dec 26 2012 CROSSREFS Different from A068108. Records give A038549, A004778, A086921. Cf. A000005, A001620, A028260, A056924, A072670, A094820, A161841, A108504. Cf. A066839, A033676, row sums of A303300. Inverse Möbius transform of A065043. Sequence in context: A345222 A327399 A122375 * A320732 A305149 A336737 Adjacent sequences: A038545 A038546 A038547 * A038549 A038550 A038551 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified February 6 13:33 EST 2023. Contains 360110 sequences. (Running on oeis4.)