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Search: id:A001055
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| A001055 |
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Number of ways of factoring n with all factors >1.
(Formerly M0095 N0032)
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+0
62
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| 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 9, 1, 5, 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) = # { k | A064553(k) = n }. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 21 2001; Benoit Cloitre and njas, May 15, 2002
Number of members of A025487 with n divisors. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 12 2004
Canfield, Erdos and Pomerance stated and Luca et al. proved that asymptotically a(n) = x^(O((log log log x)/log log x). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 07 2008
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, Amer. Math. Monthly, 102 (1995), 921-926.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].
S. R. Finch, Kalmar's composition constant
A. Murthy, Generalization of Partition Function (Introducing the Smarandache Factor Partition) [Broken link]
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Eric Weisstein's World of Mathematics, Unordered Factorization
Index entries for "core" sequences
Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function
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FORMULA
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Dirichlet g.f.: prod{n = 2 to inf}(1/(1-1/n^s)).
If n = prime^k, a(n) = partitions(k) = A000041(k).
Since A001055 (n) is the right diagonal of A066032 the given recursive formula for A066032 applies (see Maple program)
Equals A071531 * A005171 (inverse Mobius transform of the sequence "zero if prime, 1 else"). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 06 2007
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MAPLE
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with(numtheory): T := proc(n::integer, m::integer) local i, A, summe, d: if isprime(n) then: if n <= m then RETURN(1) fi: RETURN(0): fi:
A := divisors(n) minus {n, 1}: for d in A do: if d > m then A := A minus {d}: fi: od: summe := 0: for d in A do: summe := summe + T(n/d, d): od: if n <=m then summe := summe + 1: fi: RETURN(summe): end: A001055 := n -> T(n, n): [seq(A001055(n), n=1..100)];
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MATHEMATICA
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c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r]=Module[{ds, i}, ds=Select[Divisors[n], 1<#<=r&]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson Oct 28 2002 *)
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CROSSREFS
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Cf. A002033, A045778, A050322, A050336, A064553, A064554, A064555. a(p^k)=A000041. a(A002110)=A000110.
Cf. A077565.
Cf. A051731, A005171.
Sequence in context: A097203 A033273 A034836 this_sequence A129138 A112970 A112971
Adjacent sequences: A001052 A001053 A001054 this_sequence A001056 A001057 A001058
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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njas
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EXTENSIONS
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Formula and Maple program from Reinhard.Zumkeller(AT)lhsystems.com and ulrschimke(AT)aol.com
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