1,4

Comments from David Wilson (davidwwilson(AT)comcast.net), Feb 28 2009: (Start)

By a factorization of n we mean a multiset of integers > 1 whose product is n.

For example, 6 is the product of 2 such multisets, {2, 3} and {6}, so a(6) = 2.

Similarly 8 is the product of 3 such multisets, {2, 2, 2}, {2, 4} and {8}, so a(6) = 3.

1 is the product of 1 such multiset, namely the empty multiset {}, whose product is by definition the multiplicative identity 1. Hence a(1) = 1. (End)

a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 21 2001; Benoit Cloitre and N. J. A. Sloane (njas(AT)research.att.com), May 15, 2002

Number of members of A025487 with n divisors. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 12 2004

See sequence A162247 for a list of the factorizations of n and a program for generating the factorizations for any n. [From T. D. Noe (noe(AT)sspectra.com), Jun 28 2009]

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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, 1972 [alternative scanned copy].

R. E. Canfield, P. Erdos and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

R. E. Canfield, P. Erdos and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28. [A second link to the same paper.]

S. R. Finch, Kalmar's composition constant

Shamik Ghosh, Counting number of factorizations of a natural number [From T. D. Noe (noe(AT)sspectra.com), Nov 24 2008]

Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function

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

The asymptotic behavior of this sequence was studied by Canfield, Erdos and Pomerance and Luca et al. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 07 2008

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)

1: 1, a(1)=1

2: 2, a(2)=1

3: 3, a(3)=1

4: 4 = 2*2, a(4)=2

6: 6 = 2*3, a(6)=2

8: 8 = 2*4 = 2*2*2, a(8)=3

etc.

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)];

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 *)

Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Oct 29 2009: (Start)

(PARI) /* factorizations of n with factors <= m (n, m positive integers) */

fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s}

A001055(n) = fcnt(n, n) (End)

A045782 gives the range of a(n).

For records see A033833, A033834.

Cf. A002033, A045778, A050322, A050336, A064553, A064554, A064555. a(p^k)=A000041. a(A002110)=A000110.

Cf. A077565, A051731, A005171, A097296.

Sequence in context: A076526 A033273 A034836 this_sequence A129138 A112970 A112971

Adjacent sequences: A001052 A001053 A001054 this_sequence A001056 A001057 A001058

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

Formula and Maple program from Reinhard.Zumkeller(AT)lhsystems.com and ulrschimke(AT)aol.com

Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, Jun 08 2009