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A038548
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Number of divisors of n that are at most sqrt(n).
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38
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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; internal format)
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OFFSET
| 1,4
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COMMENTS
| Number of ways to arrange n identical objects in a rectangle, modulo rotation.
Number of unordered solutions of xy = n. - Colin Mallows (colinm(AT)research.avayalabs.com) 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). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 27 2008
Contribution 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))*A73668 = A038548
(End)
a(n) is the number of even divisors of 2*n that are <=sqrt(2*n) [From Joerg Arndt, Mar 04 2010]
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REFERENCES
| G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 18 Exer. 21,22
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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.
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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. [From Omar E. Pol, Jun 23 2009]
a(n) = A034178(4*n). - Michael Somos May 11 2011
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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 + ...
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MAPLE
| with(numtheory): A038548 := n->ceil(sigma[0](n)/2);
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MATHEMATICA
| Table[ Floor[ (DivisorSigma[0, n] + 1)/2], {n, 105}] [From Robert G. Wilson, v, Mar 02 2009]
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*d <= n))} /* Michael Somos Jan 25 2005 */
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CROSSREFS
| Records give A038549, A004778, A086921. Cf. A000005, A072670.
a(A025487) = A108504.
Cf. A010052, A161841, A161842. [From Omar E. Pol, Jun 23 2009]
Different from A068108.
Sequence in context: A076755 A106490 A122375 * A068108 A113309 A062362
Adjacent sequences: A038545 A038546 A038547 * A038549 A038550 A038551
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Tom Verhoeff (Tom.Verhoeff(AT)acm.org), N. J. A. Sloane (njas(AT)research.att.com).
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