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A001055
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The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).
(Formerly M0095 N0032)
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710
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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
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OFFSET
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1,4
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COMMENTS
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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(8) = 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)
See sequence A162247 for a list of the factorizations of n and a program for generating the factorizations for any n. - T. D. Noe, Jun 28 2009
So a(n) gives the number of different prime signatures that can be found among the integers that have n divisors. - Michel Marcus, Nov 11 2015
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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.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
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).
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 198, exercise 9 (in the third edition 2015, p. 296, exercise 211).
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LINKS
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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].
D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
Marc Chamberland, Colin Johnson, Alice Nadeau, and Bingxi Wu, Multiplicative Partitions, Electronic Journal of Combinatorics, 20(2) (2013), #P57.
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FORMULA
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The asymptotic behavior of this sequence was studied by Canfield, Erdős & Pomerance and Luca, Mukhopadhyay, & Srinivas. - Jonathan Vos Post, Jul 07 2008
Dirichlet g.f.: Product_{k>=2} 1/(1 - 1/k^s).
If n = p^k for a prime p, a(n) = partitions(k) = A000041(k).
Since the sequence a(n) is the right diagonal of A066032, the given recursive formula for A066032 applies (see Maple program). - Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)
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EXAMPLE
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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.
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MAPLE
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with(numtheory):
T := proc(n::integer, m::integer)
local A, summe, d:
if isprime(n) then
if n <= m then
return 1;
end if:
return 0 ;
end if:
A := divisors(n) minus {n, 1}:
for d in A do
if d > m then
A := A minus {d}:
end if:
end do:
summe := add(T(n/d, d), d=A) ;
if n <=m then
summe := summe + 1:
end if:
summe ;
end proc:
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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 *)
T[_, 1] = T[1, _] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] * T[n/#, #]&];
a[n_] := T[n, n];
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PROG
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(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}
(PARI) \\ code using Dirichlet g.f., based on Somos's code for A007896
{a(n) = my(A, v, w, m);
if(
n<1, 0,
\\ define unit vector v = [1, 0, 0, ...] of length n
v = vector(n, k, k==1);
for(k=2, n,
m = #digits(n, k) - 1;
\\ expand 1/(1-x)^k out far enough
A = (1 - x)^ -1 + x * O(x^m);
\\ w = zero vector of length n
w = vector(n);
\\ convert A to a vector
for(i=0, m, w[k^i] = polcoeff(A, i));
\\ build the answer
v = dirmul(v, w)
);
v[n]
)
};
\\ produce the sequence
(PARI) v=vector(100, k, k==1); for(n=2, #v, v+=dirmul(v, vector(#v, k, (k>1) && n^valuation(k, n)==k)) ); v \\ Max Alekseyev, Jul 16 2014
(Haskell)
a001055 = (map last a066032_tabl !!) . (subtract 1)
(Python)
from sympy import divisors, isprime
def T(n, m):
if isprime(n): return 1 if n<=m else 0
A=filter(lambda d: d<=m, divisors(n)[1:-1])
s=sum(T(n//d, d) for d in A)
return s + 1 if n<=m else s
def a(n): return T(n, n)
(Java)
public class MultiPart {
public static void main(String[] argV) {
for (int i=1; i<=100; ++i) System.out.println(1+getDivisors(2, i));
}
public static int getDivisors(int min, int n) {
int total = 0;
for (int i=min; i<n; ++i)
if (n%i==0 && n/i>=i) { ++total; if (n/i>i) total+=getDivisors(i, n/i); }
return total;
}
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CROSSREFS
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Cf. A002033, A045778, A050322, A050336, A064553, A064554, A064555, A077565, A051731, A005171, A097296, A190938, A216599, A216600, A216601, A216602.
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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EXTENSIONS
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Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, Jun 08 2009
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STATUS
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approved
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