%I #221 Jan 20 2024 09:19:10

%S 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,

%T 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,

%U 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

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

%C 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.

%C Kalmár (1931) seems to be the earliest reference that mentions this sequence (as opposed to A002033). - _N. J. A. Sloane_, May 05 2016

%C 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_, Oct 19 2005

%C 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

%C 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

%C 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 two distinct primes can be found in table A059576. - _Mats Granvik_, Sep 06 2009

%C Inverse Mobius transform of A174725 and also except for the first term, inverse Mobius transform of A174726. - _Mats Granvik_, Mar 28 2010

%C a(n) is a lower bound on the worst-case 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

%C All integers greater than 1 are perfect numbers over this sequence (for definition of A-perfect numbers, see comment to A175522). - _Vladimir Shevelev_, Aug 03 2011

%C If n is squarefree, then a(n) = A000670(A001221(n)) = A000670(A001222(n)). - _Vladimir Shevelev_ and _Franklin T. Adams-Watters_, Aug 05 2011

%C A034776 lists the values taken by this sequence. - _Robert G. Wilson v_, Jun 02 2012

%C From _Gus Wiseman_, Aug 25 2020: (Start)

%C Also the number of strict chains of divisors from n to 1. For example, the a(n) chains for n = 1, 2, 4, 6, 8, 12, 30 are:

%C 1 2/1 4/1 6/1 8/1 12/1 30/1

%C 4/2/1 6/2/1 8/2/1 12/2/1 30/2/1

%C 6/3/1 8/4/1 12/3/1 30/3/1

%C 8/4/2/1 12/4/1 30/5/1

%C 12/6/1 30/6/1

%C 12/4/2/1 30/10/1

%C 12/6/2/1 30/15/1

%C 12/6/3/1 30/6/2/1

%C 30/6/3/1

%C 30/10/2/1

%C 30/10/5/1

%C 30/15/3/1

%C 30/15/5/1

%C (End)

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

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

%D Kalmár, Laszlo, A "factorisatio numerorum" problemajarol [Hungarian], Matemat. Fizik. Lapok, 38 (1931), 1-15.

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

%H N. J. A. Sloane, <a href="/A074206/b074206.txt">Table of n, a(n) for n = 0..20000</a> (first 10000 terms from T. D. Noe)

%H Peter Brown, <a href="http://www.mountainman.com.au/harmonics.htm">Number of Ordered Factorizations</a>.

%H Peter Brown, <a href="http://www.mountainman.com.au/harmonics_01.htm">Number of Ordered Factorizations</a>.

%H Benny Chor, Paul Lemke, and Ziv Mador, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00223-X">On the number of ordered factorizations of natural numbers</a>, Discrete Math. 214 (2000), no. 1-3, 123-133. MR1743631 (2000m:11093).

%H Kristin DeVleming and Nikita Singh, <a href="https://arxiv.org/abs/2311.15922">Rational unicuspidal plane curves of low degree</a>, arXiv:2311.15922 [math.AG], 2023. See p. 14.

%H T. M. A. Fink, <a href="https://arxiv.org/abs/2307.09140">Properties of the recursive divisor function and the number of ordered factorizations</a>, arXiv:2307.09140 [math.NT], 2023.

%H E. Hille, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa215.pdf">A problem in factorisatio numerorum</a>, Acta Arith., 2 (1936), 134-144.

%H E. Hille, <a href="https://doi.org/10.1215/S0012-7094-37-00344-2">The inversion problem of Möbius</a>, Duke Math. J., 3 (1937), 549-568.

%H Shikao Ikehara, <a href="https://doi.org/10.11429/ppmsj1919.21.5_208">On Kalmar's Problem in "Factorisatio Numerorum"</a>, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208-219.

%H Shikao Ikehara, <a href="https://doi.org/10.11429/ppmsj1919.23.0_767">On Kalmar's Problem in "Factorisatio Numerorum" II</a>, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767-774.

%H Laszlo Kalmár, <a href="http://pub.acta.hu/acta/showCustomerArticle.action?id=5296&amp;dataObjectType=article">Über die mittlere Anzahl der Produktdarstellungen der Zahlen. (Erste Mitteilung)</a>, Acta Litt. ac Scient. Szeged 5 (1931): 95-107.

%H M. Klazar and F. Luca, <a href="https://arxiv.org/abs/math/0505352">On the maximal order of numbers in the "factorisatio numerorum" problem</a>, arXiv:math/0505352 [math.NT], 2005-2006.

%H Arnold Knopfmacher and Michael Mays, <a href="http://www.mathematica-journal.com/issue/v10i1/Factorizations.html">Ordered and Unordered Factorization of Integers</a>, The Mathematica Journal, Volume 10, Issue 1 p. 72.

%H Arnau Mir, Francesc Rossello, and Lucia Rotger, <a href="https://arxiv.org/abs/1805.01329">Sound Colless-like balance indices for multifurcating trees</a>, arXiv:1805.01329 [q-bio.PE], 2018.

%H Augustine O. Munagi, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r50">Labeled factorization of integers</a>, INTEGERS: The Electronic Journal of Combinatorics 16:1 (2009), #R50.

%H L. A. Newberg and D. Naor, <a href="http://dx.doi.org/10.1006/aama.1993.1009">A lower bound on the number of solutions to the probed partial digest problem</a>, Advances in Applied Mathematics, 14(2), 1993, 172-183. doi: 10.1006/aama.1993.1009.

%H Ray Tomes, <a href="http://ray.tomes.biz/maths.html">The Maths and Physics of the Harmonics Theory</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPartition.html">Perfect Partition</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OrderedFactorization.html">Ordered Factorization</a>.

%H David W. Wilson, <a href="/A074206/a074206.txt">Comments on A074206 and related sequences</a>.

%H David W. Wilson, <a href="/A074206/a074206.pl.txt">Perl program for A074206</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F With different offset: a(n) = sum of all a(i) such that i divides n and i < n. - _Clark Kimberling_

%F a(p^k) = 2^(k-1) if k>0 and p is a prime.

%F Dirichlet g.f.: 1/(2-zeta(s)). - Herbert S. Wilf, Apr 29 2003

%F a(n) = A067824(n)/2 for n>1; a(A122408(n)) = A122408(n)/2. - _Reinhard Zumkeller_, Sep 03 2006

%F 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]

%F a(0) = 0, a(1) = 1; a(n) = [x^n] Sum_{k=1..n-1} a(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Dec 11 2017

%F a(n) = a(A046523(n)); a(A025487(n)) = A050324(n): a(n) depends only on the nonzero exponents in the prime factorization of n, more precisely prime signature of n, cf. A124010 and A320390. - _M. F. Hasler_, Oct 12 2018

%F a(n) = A000670(A001221(n)) for squarefree n. In particular a(A002110(n)) = A000670(n). - _Amiram Eldar_, May 13 2019

%F a(n) = A050369(n)/n, for n>=1. - _Ridouane Oudra_, Aug 31 2019

%F a(n) = A361665(A181819(n)). - _Pontus von Brömssen_, Mar 25 2023

%F From _Ridouane Oudra_, Nov 02 2023: (Start)

%F If p,q are distinct primes, and n,m>0 then we have:

%F a(p^n*q^m) = Sum_{k=0..min(n,m)} 2^(n+m-k-1)*binomial(n,k)*binomial(m,k);

%F More generally: let tau[k](n) denote the number of ordered factorizations of n as a product of k terms, also named the k-th Piltz function (see A007425), then we have for n>1:

%F a(n) = Sum_{j=1..bigomega(n)} Sum_{k=1..j} (-1)^(j-k)*binomial(j,k)*tau[k](n), or

%F a(n) = Sum_{j=1..bigomega(n)} Sum_{k=0..j-1} (-1)^k*binomial(j,k)*tau[j-k](n). (End)

%e 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 + ...

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

%p 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: # _James A. Sellers_, Dec 07 2000

%p A074206:= proc(n) option remember; if n > 1 then `+`(op(apply(A074206, numtheory[divisors](n)[1..-2]))) else n fi end: # _M. F. Hasler_, Oct 12 2018

%t a[0] = 0; 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 *)

%t ccc[n_]:=Switch[n,0,{},1,{{1}},_,Join@@Table[Prepend[#,n]&/@ccc[d],{d,Most[Divisors[n]]}]]; Table[Length[ccc[n]],{n,0,100}] (* _Gus Wiseman_, Aug 25 2020 *)

%o (Haskell)

%o a074206 n | n <= 1 = n

%o | otherwise = 1 + (sum $ map (a074206 . (div n)) $

%o tail $ a027751_row n)

%o -- _Reinhard Zumkeller_, Oct 01 2012

%o (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

%o (PARI) {a(n) = if( n<2, n>0, my(A = divisors(n)); sum(k=1, #A-1, a(A[k])))}; /* _Michael Somos_, Dec 26 2016 */

%o (PARI) A074206(n)=if(n>1, sumdiv(n, i, if(i<n, A074206(i))),n) \\ _M. F. Hasler_, Oct 12 2018

%o (PARI) A74206=[1]; A074206(n)={if(#A74206<n, A74206=concat(A74206,vector(n*3\/2-#A74206)), n&& A74206[n], return(A74206[n])); A74206[n]=sumdiv(n, i, if(i<4, i<n, i<n, A074206(i)))} \\ Use memoization for computing many values. - _M. F. Hasler_, Oct 12 2018

%o (PARI) first(n) = {my(res = vector(n, i, 1)); for(i = 2, n, for(j = 2, n \ i, res[i*j] += res[i])); concat(0, res)} \\ _David A. Corneth_, Oct 13 2018

%o (PARI) first(n) = {my(res = vector(n, i, 1)); for(i = 2, n, d = divisors(i); res[i] += sum(j = 1, #d-1, res[d[j]])); concat(0, res)} \\ somewhat faster than progs above for finding first terms of n. \\ _David A. Corneth_, Oct 12 2018

%o (PARI) a(n)={if(!n, 0, my(sig=factor(n)[,2], m=vecsum(sig)); sum(k=0, m, prod(i=1, #sig, binomial(sig[i]+k-1, k-1))*sum(r=k, m, binomial(r,k)*(-1)^(r-k))))} \\ _Andrew Howroyd_, Aug 30 2020

%o (SageMath)

%o @cached_function

%o def minus_mu(n):

%o if n < 2: return n

%o return sum(minus_mu(d) for d in divisors(n)[:-1])

%o # Note that changing the sign of the sum gives the Möbius function A008683.

%o print([minus_mu(n) for n in (0..96)]) # _Peter Luschny_, Dec 26 2016

%o (Python)

%o from math import prod

%o from functools import lru_cache

%o from sympy import divisors, factorint, prime

%o @lru_cache(maxsize=None)

%o def A074206(n): return sum(A074206(d) for d in divisors(prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n).values(),reverse=True))),generator=True,proper=True)) if n > 1 else n # _Chai Wah Wu_, Sep 16 2022

%Y Apart from initial term, same as A002033.

%Y a(A002110) = A000670, row sums of A251683.

%Y A173382 (and A025523) gives partial sums.

%Y Cf. A008683, A050324, A027751, A001221, A034776, A000670, A050369.

%Y A124433 has these as unsigned row sums.

%Y A334996 has these as row sums.

%Y A001055 counts factorizations.

%Y A001222 counts prime factors with multiplicity.

%Y A008480 counts ordered prime factorizations.

%Y A067824 counts strict chains of divisors starting with n.

%Y A122651 counts strict chains of divisors summing to n.

%Y A253249 counts strict chains of divisors.

%Y Cf. A000005, A025487, A046523, A059576, A122408, A124010, A167865, A174725, A174726, A175522, A181819, A320390, A334997, A337105, A361665.

%K nonn,core,easy,nice

%O 0,5

%A _N. J. A. Sloane_, Apr 29 2003

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