

A074206


Kalmár's [Kalmar's] problem: number of ordered factorizations of n.


35



0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 44, 1, 3, 8, 32, 3, 13, 1, 8, 3, 13, 1, 76, 1, 3, 8, 8, 3, 13, 1, 48, 8, 3, 1, 44, 3, 3, 3, 20, 1, 44, 3, 8, 3, 3, 3, 112
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OFFSET

0,5


COMMENTS

a(0)=0, a(1)=1; thereafter a(n) is the number of ordered factorizations of n as a product of integers greater than 1.
Kalmár (1931) seems to be the earliest reference that mentions this sequence (as opposed to A002033).  N. J. A. Sloane, May 05 2016
a(n) is the permanent of the n1 X n1 matrix A with (i,j) entry = 1 if ji+1 and = 0 otherwise. This is because ordered factorizations correspond to nonzero elementary products in the permanent. For example, with n=6, 3*2 > 1,3,6 [partial products] > 6,3,1 [reverse list] > (6,3)(3,1) [partition into pairs with offset 1] > (5,3)(2,1) [decrement first entry] > (5,3)(2,1)(1,2)(3,4)(4,5) [append pairs (i,i+1) to get a permutation] > elementary product A(1,2)A(2,1)A(3,4)A(4,5)A(5,3).  David Callan, Oct 19 2005
This sequence is important in describing the amount of energy in all wave structures in the Universe according to harmonics theory.  Ray Tomes (ray(AT)tomes.biz), Jul 22 2007
a(n) appears to be the number of permutation matrices contributing to the Moebius function. See A008683 for more information. Also a(n) appears to be the Moebius transform of A067824. Furthermore it appears that except for the first term a(n)=A067824(n)*(1/2). Are there other sequences such that when the Moebius transform is applied, the new sequence is also a constant factor times the starting sequence?  Mats Granvik, Jan 01 2009
Numbers divisible by n distinct primes appear to have ordered factorization values that can be found in an ndimensional summatory Pascal triangle. For example, the ordered factorization values for numbers divisible by two distinct primes can be found in table A059576.  Mats Granvik, Sep 06 2009
Inverse Mobius transform of A174725 and also except for the first term, inverse Mobius transform of A174726.  Mats Granvik, Mar 28 2010
a(n) is a lower bound on the worstcase number of solutions to the probed partial digest problem for n fragments of DNA; see the Newberg & Naor reference, below.  Lee A. Newberg, Aug 02 2011
All integers greater than 1 are perfect numbers over this sequence (for definition of Aperfect numbers, see comment to A175522).  Vladimir Shevelev, Aug 03 2011
If n is squarefree, then a(n) = A000670(A001221(n)) = A000670(A001222(n)).  Vladimir Shevelev and Franklin T. AdamsWatters, Aug 05 2011
A034776 lists the values taken by this sequence.  Robert G. Wilson v, Jun 02 2012


REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.
Kalmár, Laszlo, A "factorisatio numerorum" problemajarol [Hungarian], Matemat. Fizik. Lapok, 38 (1931), 115.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.


LINKS

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 0..20000, May 05 2016 [First 10000 terms from T. D. Noe]
Peter Brown, Number of Ordered Factorizations
Peter Brown, Number of Ordered Factorizations
Benny Chor, Paul Lemke, Ziv Mador, On the number of ordered factorizations of natural numbers, Discrete Math. 214 (2000), no. 13, 123133. MR1743631 (2000m:11093).
E. Hille, A problem in factorisatio numerorum, Acta Arith., 2 (1936), 134144.
E. Hille, The inversion problem of Möbius, Duke Math. J., 3 (1937), 549568.
Shikao Ikehara, On Kalmar's Problem in "Factorisatio Numerorum", Proceedings of the PhysicoMathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208219.
Shikao Ikehara, On Kalmar's Problem in "Factorisatio Numerorum" II, Proceedings of the PhysicoMathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767774.
Laszlo Kalmár, Über die mittlere Anzahl der Produktdarstellungen der Zahlen. (Erste Mitteilung), Acta Litt. ac Scient. Szeged 5 (1931): 95107.
M. Klazar and F. Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, arXiv:math/0505352 [math.NT], 20052006.
Arnold Knopfmacher & Michael Mays, Ordered and Unordered Factorization of Integers, The Mathematica Journal, Volume 10, Issue 1 p. 72.
Augustine O. Munagi, Labeled factorization of integers, INTEGERS: The Electronic Journal of Conminatorics 16:1 (2009), #R50.
L. A. Newberg & D. Naor, A lower bound on the number of solutions to the probed partial digest problem, Advances in Applied Mathematics, 14(2), 1993, 172183. doi: 10.1006/aama.1993.1009
Ray Tomes, The Maths and Physics of the Harmonics Theory
Eric Weisstein's World of Mathematics, Perfect Partition
Eric Weisstein's World of Mathematics, Ordered Factorization
David W. Wilson, Comments on A074206 and related sequences
David W. Wilson, Perl program for A074206
Index entries for "core" sequences


FORMULA

With different offset: a(n) = sum of all a(i) such that i divides n and i < n.  Clark Kimberling
a(p^k) = 2^(k1) if k>0 and p is a prime.
Dirichlet g.f.: 1/(2zeta(s)).  Herbert S. Wilf, Apr 29 2003
a(n) = A067824(n)/2 for n>1; a(A122408(n)) = A122408(n)/2.  Reinhard Zumkeller, Sep 03 2006
If p,q,r,... are distinct primes, then a(p*q)=3, a(p^2*q)=8, a(p*q*r)=13, a(p^3*q)=20, etc.  Vladimir Shevelev, Aug 03 2011 [corrected by Charles R Greathouse IV, Jun 02 2012]
a(0) = 0, a(1) = 1; a(n) = [x^n] Sum_{k=1..n1} a(k)*x^k/(1  x^k).  Ilya Gutkovskiy, Dec 11 2017


EXAMPLE

G.f. = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 3*x^10 + ...
Number of ordered factorizations of 8 is 4: 8 = 2*4 = 4*2 = 2*2*2.


MAPLE

a := array(1..150): for k from 1 to 150 do a[k] := 0 od: a[1] := 1: for j from 2 to 150 do for m from 1 to j1 do if j mod m = 0 then a[j] := a[j]+a[m] fi: od: od: for k from 1 to 150 do printf(`%d, `, a[k]) od: # James A. Sellers, Dec 07 2000


MATHEMATICA

a[0] = 1; a[1] = 1; a[n_] := a[n] = a /@ Most[Divisors[n]] // Total; a /@ Range[20000] (* N. J. A. Sloane, May 04 2016, based on program in A002033 *)


PROG

(Haskell)
a074206 n  n <= 1 = n
 otherwise = 1 + (sum $ map (a074206 . (div n)) $
tail $ a027751_row n)
 Reinhard Zumkeller, Oct 01 2012
(PARI) A=vector(100); A[1]=1; for(n=2, #A, A[n]=1+sumdiv(n, d, A[d])); A/=2; A[1]=1; concat(0, A) \\ Charles R Greathouse IV, Nov 20 2012
(Sage)
@cached_function
def minus_mu(n):
if n < 2 : return n
return sum(minus_mu(d) for d in divisors(n)[:1])
# Note that changing the sign of the sum gives the Möbius function A008683.
print [minus_mu(n) for n in (0..96)] # Peter Luschny, Dec 26 2016
(PARI) {a(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A1, a(A[k])))}; /* Michael Somos, Dec 26 2016 */


CROSSREFS

Apart from initial term, same as A002033.
a(A002110) = A000670, row sums of A251683.
A173382 (and A025523) gives partial sums.
Cf. A001055, A008683, A050324, A027751, A001221, A001222, A034776.
Sequence in context: A296119 A118314 A002033 * A173801 A108466 A211110
Adjacent sequences: A074203 A074204 A074205 * A074207 A074208 A074209


KEYWORD

nonn,core,easy,nice


AUTHOR

N. J. A. Sloane, Apr 29 2003


EXTENSIONS

Originally this sequence was merged with A002033, the number of perfect partitions. Herbert S. Wilf suggested that it warrants an entry of its own.


STATUS

approved



