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A027423
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Number of positive divisors of n!.
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48
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1, 1, 2, 4, 8, 16, 30, 60, 96, 160, 270, 540, 792, 1584, 2592, 4032, 5376, 10752, 14688, 29376, 41040, 60800, 96000, 192000, 242880, 340032, 532224, 677376, 917280, 1834560, 2332800, 4665600, 5529600, 7864320, 12165120, 16422912
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| It appears that a(n+1)=2*a(n) if n is in A068499. - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 07 2002
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
D. Berend et al., Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.
Paul Erdos, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the Number of Divisors of n!, in Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand, Birkhauser 1996, pp. 337-355.
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FORMULA
| a(n) <= a(n+1) <= 2*a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 07 2002
Contribution from Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 28 2009: (Start)
Assume, p1,p2...pm are the prime numbers less or equal to n.
bk=sum_{i=1,2,3...} floor(n/pk^i)
Then, a(n)= product_{i=1,2...,m} (bi+1)
For example, if n=5, p1=2,p2=3,p3=5
b1=floor(5/2)+floor(5/2^2)+floor(5/2^3)+...=2+1+0+..=3 similarly, b2=b3=1
Thus a(5)=(3+1)(1+1)(1+1)=16 (End)
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EXAMPLE
| a(4) = 8 because 4!=24 has precisely eight distinct divisors: 1,2,3,4,6,8,12,24.
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MAPLE
| A027423 := n -> numtheory[tau](n!);
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MATHEMATICA
| Table[ DivisorSigma[0, n! ], {n, 0, 35}]
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PROG
| (PARI) for(k=0, 50, print1(numdiv(k!), ", ")) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 09 2009]
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CROSSREFS
| Sequence in context: A164259 A164203 A164178 * A140410 A164263 A018763
Adjacent sequences: A027420 A027421 A027422 * A027424 A027425 A027426
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Glen Burch (gburch(AT)erols.com), Leroy Quet.
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