0,5

a(n) is the permanent of the n-1 X n-1 matrix A with (i,j) entry = 1 if j|i+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 (callan(AT)stat.wisc.edu), 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

Contribution from Mats Granvik (mats.granvik(AT)abo.fi), Jan 01 2009: (Start)

a(n) appears to be the number of permutation matrices contributing to the Moebius function. See A008683 for more information.

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 factor times the starting sequence? (End)

Numbers divisible by n distinct primes appear to have ordered factorization values that can be found in an n-dimensional summatory Pascal triangle. For example, the ordered factorization values for numbers divisible by 2 distinct primes can be found in table A059576. [From Mats Granvik (mats.granvik(AT)abo.fi), Sep 06 2009]

If A002033(n)=A074206(n+1) and A074206=(zero together with number of perfect partiions of n), then A074206=(number of ordered factorizations of n-1)? [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Nov 22 2009]

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.

R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.

T. D. Noe, Table of n, a(n) for n = 0..10000

Peter Brown, Title?

Peter Brown, Title?

M. Klazar and F. Luca, On the maximal order of numbers in the "factorisatio numerorum" problem

Ray Tomes, The Maths and Physics of the Harmonics Theory

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

With different offset: a(n) = sum of all a(i) such that i divides n and i < n (Clark Kimberling).

a(p^k)=2^(k-1).

Dirichlet g.f.: 1/(2-zeta(s)). - Herb Wilf, Apr 29, 2003

a(n) = A067824(n)/2 for n>1; a(A122408(n)) = A122408(n)/2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 03 2006

Number of ordered factorizations of 8 is 4: 8 = 2*4 = 4*2 = 2*2*2.

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 j-1 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: # from James A. Sellers Dec 07 2000

Apart from initial term, same as A002033. Cf. A001055, A050324. a(A002110)=A000670.

Sequence in context: A097283 A118314 A002033 this_sequence A108466 A087145 A117172

Adjacent sequences: A074203 A074204 A074205 this_sequence A074207 A074208 A074209

nonn,core,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Apr 29, 2003

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