

A038548


Number of divisors of n that are at most sqrt(n).


69



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
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OFFSET

1,4


COMMENTS

Number of ways to arrange n identical objects in a rectangle, modulo rotation.
Number of unordered solutions of xy = n.  Colin Mallows, Jan 26 2002
Number of ways to write n1 as n1 = x*y + x + y, 0<=x<=y<=n.  Benoit Cloitre, Jun 23 2002
Also number of values for x where x+2n and x2n are both squares (e.g., if n=9, then 18+18 and 1818 are both squares, as are 82+18 and 8218 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 x2n=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,...,k1 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^2y^2=16 n. For example, a(24)=4 because there are 4 solutions: (x,y)=(10,4),(11,10),(14,20),(25,46).  NE. 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_k2)/(k+2) or equivalently by x_k = n/(k+1)  k.  Luc Rousseau, Mar 03 2018


REFERENCES

G. E. Andrews, K. 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, Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, 2016, vol. 170, pp. 1734.
Christopher Briggs, Y. Hirano, H. Tsutsui, Positive Solutions to Some Systems of Diophantine Equations, Journal of Integer Sequences, 2016 Vol 19 #16.8.4.
S.H. Cha, E. G. DuCasse, L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT], 2014, (2.27).
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #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)/(1x^k)).  Jon Perry, Sep 10 2004
Dirichlet g.f.: (zeta(s) + zeta(2*s))/2.  Christian G. Bower, Jun 06 2005
a(n) = (A000005(n) + A010052(n))/2.  Omar E. Pol, Jun 23 2009
a(n) = A034178(4*n).  Michael Somos, May 11 2011


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[0](n)/2);


MATHEMATICA

Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] (* Robert G. Wilson v, Mar 02 2009 *)


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, A072670, A094820, A161841, A108504.
Cf. A066839, A033676.
Sequence in context: A317751 A106490 A122375 * A320732 A305149 A068108
Adjacent sequences: A038545 A038546 A038547 * A038549 A038550 A038551


KEYWORD

nonn,easy,nice


AUTHOR

Tom Verhoeff, N. J. A. Sloane


STATUS

approved



