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A038548 Number of divisors of n that are at most sqrt(n). 57
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 xy = 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

From Eric Desbiaux, Nov 16 2008: (Start)

Sum(n=1...inf) ((10^(-(n)))^(n))*(10^n)/(10^n-1)

=Sum(n=1...inf)((10^(-n))^((-1)^n))*(1/(10^n-1))

=Sum(n=1...inf)((10^(-n))^((-1)^n))*A073668 = A038548

(End)

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

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

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

FORMULA

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) + 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: A076755 A106490 A122375 * A068108 A113309 A062362

Adjacent sequences:  A038545 A038546 A038547 * A038549 A038550 A038551

KEYWORD

nonn,easy,nice

AUTHOR

Tom Verhoeff (Tom.Verhoeff(AT)acm.org), N. J. A. Sloane

STATUS

approved

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Last modified September 2 10:02 EDT 2014. Contains 246348 sequences.