Search: id:A001055
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%I A001055 M0095 N0032
%S A001055 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,
%T A001055 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,
%U A001055 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
%N A001055 Number of ways of factoring n with all factors greater than 1 (a(1)=1 
               by convention).
%C A001055 Comments from David Wilson (davidwwilson(AT)comcast.net), Feb 28 2009: 
               (Start)
%C A001055 By a factorization of n we mean a multiset of integers > 1 whose product 
               is n.
%C A001055 For example, 6 is the product of 2 such multisets, {2, 3} and {6}, so 
               a(6) = 2.
%C A001055 Similarly 8 is the product of 3 such multisets, {2, 2, 2}, {2, 4} and 
               {8}, so a(6) = 3.
%C A001055 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)
%C A001055 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
%C A001055 Number of members of A025487 with n divisors. - Matthew Vandermast (ghodges14(AT)comcast.net), 
               Jul 12 2004
%C A001055 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]
%D A001055 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 A001055 D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
%D A001055 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
%D A001055 R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, 
               Amer. Math. Monthly, 102 (1995), 921-926.
%D A001055 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.
%D A001055 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001055 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001055 T. D. Noe, <a href="b001055.txt">Table of n, a(n) for n = 1..10000</a>
%H A001055 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A001055 R. E. Canfield, P. Erdos and C. Pomerance, <a href="http://math.dartmouth.edu/
               ~carlp/PDF/paper39.pdf">On a Problem of Oppenheim concerning "Factorisatio 
               Numerorum"</a>, J. Number Theory 17 (1983), 1-28.
%H A001055 R. E. Canfield, P. Erdos and C. Pomerance, <a href="http://renyi.hu/~p_erdos/
               1983-11.pdf">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</
               a>, J. Number Theory 17 (1983), 1-28. [A second link to the same 
               paper.]
%H A001055 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Kalmar's composition 
               constant</a>
%H A001055 Shamik Ghosh, <a href="http://arxiv.org/abs/0811.3479">Counting number 
               of factorizations of a natural number</a> [From T. D. Noe (noe(AT)sspectra.com), 
               Nov 24 2008]
%H A001055 Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, <a href="http:/
               /arxiv.org/pdf/0807.0986">On the Oppenheim's "factorisatio numerorum" 
               function</a>
%H A001055 A. Murthy, <a href="http://www.gallup.unm.edu/~smarandache/murthy11.htm">
               Generalization of Partition Function (Introducing the Smarandache 
               Factor Partition)</a> [Broken link]
%H A001055 Amarnath Murthy and Charles Ashbacher, <a href="http://www.gallup.unm.edu/
               ~smarandache/MurthyBook.pdf">Generalized Partitions and Some New 
               Ideas on Number Theory and Smarandache Sequences</a>, Hexis, Phoenix; 
               USA 2005. See Section 1.4.
%H A001055 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               UnorderedFactorization.html">Unordered Factorization</a>
%H A001055 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A001055 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
%F A001055 Dirichlet g.f.: prod{n = 2 to inf}(1/(1-1/n^s)).
%F A001055 If n = prime^k, a(n) = partitions(k) = A000041(k).
%F A001055 Since A001055 (n) is the right diagonal of A066032 the given recursive 
               formula for A066032 applies (see Maple program)
%e A001055 1: 1, a(1)=1
%e A001055 2: 2, a(2)=1
%e A001055 3: 3, a(3)=1
%e A001055 4: 4 = 2*2, a(4)=2
%e A001055 6: 6 = 2*3, a(6)=2
%e A001055 8: 8 = 2*4 = 2*2*2, a(8)=3
%e A001055 etc.
%p A001055 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:
%p A001055 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)];
%t A001055 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 *)
%o A001055 Contribution from Michael Porter (michael_b_porter(AT)yahoo.com), Oct 
               29 2009: (Start)
%o A001055 (PARI) /* factorizations of n with factors <= m (n,m positive integers) 
               */
%o A001055 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}
%o A001055 A001055(n) = fcnt(n,n) (End)
%Y A001055 A045782 gives the range of a(n).
%Y A001055 For records see A033833, A033834.
%Y A001055 Cf. A002033, A045778, A050322, A050336, A064553, A064554, A064555. a(p^k)=A000041. 
               a(A002110)=A000110.
%Y A001055 Cf. A077565, A051731, A005171, A097296.
%Y A001055 Sequence in context: A076526 A033273 A034836 this_sequence A129138 A112970 
               A112971
%Y A001055 Adjacent sequences: A001052 A001053 A001054 this_sequence A001056 A001057 
               A001058
%K A001055 nonn,easy,nice,core
%O A001055 1,4
%A A001055 N. J. A. Sloane (njas(AT)research.att.com).
%E A001055 Formula and Maple program from Reinhard.Zumkeller(AT)lhsystems.com and 
               ulrschimke(AT)aol.com
%E A001055 Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, 
               Jun 08 2009

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