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 A050363 Number of ordered factorizations into prime powers greater than 1. 4
 1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 5, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 6, 1, 16, 2, 2, 2, 14, 1, 2, 2, 12, 1, 6, 1, 5, 5, 2, 1, 28, 2, 5, 2, 5, 1, 12, 2, 12, 2, 2, 1, 18, 1, 2, 5, 32, 2, 6, 1, 5, 2, 6, 1, 37, 1, 2, 5, 5, 2, 6, 1, 28, 8, 2, 1, 18, 2, 2, 2, 12, 1, 18, 2, 5, 2, 2, 2, 64 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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). The Dirichlet inverse is in A010055, turning all but the first element in A010055 negative. - R. J. Mathar, Jul 15 2010 Not multiplicative: a(6) =2 <> a(2)*a(3) = 1*1. - R. J. Mathar, May 25 2017 LINKS R. J. Mathar, Table of n, a(n) for n = 1..5000 FORMULA Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of prime powers >1. a(p^k) = 2^(k-1). a(A002110(k)) = k!. a(n) = A050364(A101296(n)). - R. J. Mathar, May 26 2017 G.f. A(x) satisfies: A(x) = x + Sum_{p prime, k>=1} A(x^(p^k)). - Ilya Gutkovskiy, May 11 2019 EXAMPLE From R. J. Mathar, May 25 2017: (Start) a(p^2)  = 2: factorizations p^2, p*p. a(p^3)  = 4: factorizations p^3, p^2*p, p*p^2, p*p*p. a(p*q)  = 2: factorizations p*q, q*p. a(p*q^2)= 5: factorizations p*q^2, q^2*p, p*q*q, q*p*q, q*q*p. (End) MAPLE read(transforms) ; [1, seq(-A010055(n), n=2..100)] ; DIRICHLETi(%) ; # R. J. Mathar, May 25 2017 CROSSREFS Cf. A000961, A050360, A050361, A050362, A050363, A050364, A076277. Sequence in context: A077191 A317545 A319002 * A308063 A166974 A292504 Adjacent sequences:  A050360 A050361 A050362 * A050364 A050365 A050366 KEYWORD nonn AUTHOR Christian G. Bower, Oct 15 1999 STATUS approved

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Last modified September 28 09:32 EDT 2021. Contains 347714 sequences. (Running on oeis4.)