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Search: id:A046951
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| A046951 |
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a(n) = |{(i,j):i*j=n AND i|j}| = |{(i,j):i*j^2=n}|. Also tau(A000188); also number of squares dividing n. |
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+0
37
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| 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 4, 1, 1, 1, 2, 1
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Invented by the HR automatic theory formation program.
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).
a(A130279(n))=n and a(m)<>n for m<A130279(n); A008966(n)=0^(a(n)-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2007
sum_{k=1;inf} 1/(10^(k^2)-1) [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Mar 24 2009]
We have a(n) = A159631(n) for all n<125, but a(125)=2<3=A159631(125). [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 22 2009]
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LINKS
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R. Zumkeller, Table of n, a(n) for n = 1..10000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics
N. J. A. Sloane, Transforms
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FORMULA
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Multiplicative with p^e --> floor(e/2)+1, p prime. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2007
Inverse Moebius transform of characteristic function of squares. Dirichlet g.f.: zeta(s)*zeta(2s).
First differences of A013936. Average value tends towards pi^2/6=1.644934... (A013661, A013679). - Henry Bottomley (se16(AT)btinternet.com), Aug 16 2001
G.f.: Sum_{k>0} x^(k^2)/(1-x^(k^2)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 13 2002
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EXAMPLE
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f(16) = 3 because 1*16=16 and 1|16, 2*8=16 and 2|8, 4*4=16 and 4|4.
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CROSSREFS
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Cf. A000188, A004101, A005117, A038538, A046952, A052304. a(p^k)=A008619=[n/2]+1. a(A002110)=1.
Cf. A159631 [From S. R. Finch (Steven.Finch(AT)inria.fr), Apr 22 2009]
Sequence in context: A088737 A096309 A049419 this_sequence A159631 A050377 A001826
Adjacent sequences: A046948 A046949 A046950 this_sequence A046952 A046953 A046954
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KEYWORD
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nice,nonn,mult
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AUTHOR
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Simon Colton (simonco(AT)cs.york.ac.uk)
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