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Search: id:A005361
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| A005361 |
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Product of exponents of prime factorization of n.
(Formerly M0063)
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+0
19
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| 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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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).
There was an old comment here that said "a(n) is the number of nilpotents elements in the ring Z/nZ", but this is false - see A003557.
a(n) is the number of square-full divisors of n. a(n) is also the number of divisors d of n such that d and n have the same prime factors, i.e., A007947(d)=A007947(n). [From Laszlo Toth (ltoth(AT)ttk.pte.hu), May 22 2009]
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REFERENCES
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J. Knopfmacher, A prime-divisor function, Proc. Amer. Math. Soc., 40 (1973), 373-377. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), May 22 2009]
Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc., 34 (1972), 79-80. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), May 22 2009]
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..100000
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FORMULA
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n = Product (p_j^k_j) -> a(n) = Product (k_j). Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).
Multiplicative with a(p^e) = e. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
a(n)=Sum(d dividing n, floor(rad(d)/rad(n)), where rad(n) is A007947 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Nov 06 2009]
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MATHEMATICA
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Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
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PROGRAM
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(PARI) for(n=1, 100, print1(prod(i=1, omega(n), component(component(factor(n), 2), i)), ", "))
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CROSSREFS
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Cf. A000005, A052306. a(p^k)=A000027=n. a(A002110)=A000012=1.
Sequence in context: A157754 A072411 A091050 this_sequence A008479 A107345 A000688
Adjacent sequences: A005358 A005359 A005360 this_sequence A005362 A005363 A005364
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KEYWORD
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nonn,easy,nice,mult,new
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AUTHOR
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Jeffrey Shallit, Olivier Gerard (olivier.gerard(AT)gmail.com)
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