A067824 -id:A067824 - cp's OEIS frontend

A002033 Number of perfect partitions of n.

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

Offset: 0

N. J. A. Sloane

A perfect partition of n is one which contains just one partition of every number less than n when repeated parts are regarded as indistinguishable. Thus 1^n is a perfect partition for every n; and for n = 7, 4 1^3, 4 2 1, 2^3 1 and 1^7 are all perfect partitions. [Riordan]
Also number of ordered factorizations of n+1, see A074206.
Also number of gozinta chains from 1 to n (see A034776). - David W. Wilson
a(n) is the permanent of the n X n matrix with (i,j) entry = 1 if j|i+1 and = 0 otherwise. For n=3 the matrix is {{1, 1, 0}, {1, 0, 1}, {1, 1, 0}} with permanent = 2. - David Callan, Oct 19 2005
Appears to be the number of permutations that contribute to the determinant that gives the Moebius function. Verified up to a(9). - Mats Granvik, Sep 13 2008
Dirichlet inverse of A153881 (assuming offset 1). - Mats Granvik, Jan 03 2009
Equals row sums of triangle A176917. - Gary W. Adamson, Apr 28 2010
A partition is perfect iff it is complete (A126796) and knapsack (A108917). - Gus Wiseman, Jun 22 2016
a(n) is also the number of series-reduced planted achiral trees with n + 1 unlabeled leaves, where a rooted tree is series-reduced if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. Also Moebius transform of A067824. - Gus Wiseman, Jul 13 2018

			n=0: 1 (the empty partition)
n=1: 1 (1)
n=2: 1 (11)
n=3: 2 (21, 111)
n=4: 1 (1111)
n=5: 3 (311, 221, 11111)
n=6: 1 (111111)
n=7: 4 (4111, 421, 2221, 1111111)
From _Gus Wiseman_, Jul 13 2018: (Start)
The a(11) = 8 series-reduced planted achiral trees with 12 unlabeled leaves:
  (oooooooooooo)
  ((oooooo)(oooooo))
  ((oooo)(oooo)(oooo))
  ((ooo)(ooo)(ooo)(ooo))
  ((oo)(oo)(oo)(oo)(oo)(oo))
  (((ooo)(ooo))((ooo)(ooo)))
  (((oo)(oo)(oo))((oo)(oo)(oo)))
  (((oo)(oo))((oo)(oo))((oo)(oo)))
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.
D. E. Knuth, The Art of Computer Programming, Pre-Fasc. 3b, Sect. 7.2.1.5, no. 67, p. 25.
P. A. MacMahon, The theory of perfect partitions and the compositions of multipartite numbers, Messenger Math., 20 (1891), 103-119.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 123-124.
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).

T. D. Noe, Table of n, a(n) for n = 0..9999
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
HoKyu Lee, Double perfect partitions, Discrete Math., 306 (2006), 519-525.
Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1970
Eric Weisstein's World of Mathematics, Perfect Partition
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Index entries for "core" sequences

Same as A074206, up to the offset and initial term there.
Cf. A001055, A050324.
a(A002110)=A000670.
Cf. A000123, A100529, A117621.
Cf. A176917.
For parity see A008966.
Cf. A126796, A108917.
Cf. A001678, A003238, A067824, A167865, A214577, A289078, A292504, A316782.

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 Sellers, Dec 07 2000
# alternative
A002033 := proc(n)
    option remember;
    local a;
    if n <= 2 then
        return 1;
    else
        a := 0 ;
        for i from 0 to n-1 do
            if modp(n+1,i+1) = 0 then
                a := a+procname(i);
            end if;
        end do:
    end if;
    a ;
end proc: # R. J. Mathar, May 25 2017

a[0] = 1; a[1] = 1; a[n_] := a[n] = a /@ Most[Divisors[n]] // Total; a /@ Range[96]  (* Jean-François Alcover, Apr 06 2011, updated Sep 23 2014. NOTE: This produces A074206(n) = a(n-1). - M. F. Hasler, Oct 12 2018 *)

A002033(n) = if(n,sumdiv(n+1,i,if(i<=n,A002033(i-1))),1) \\ Michael B. Porter, Nov 01 2009, corrected by M. F. Hasler, Oct 12 2018

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A002033(n):
    if n <= 1:
        return 1
    return sum(A002033(i-1) for i in divisors(n+1,generator=True) if i <= n) # Chai Wah Wu, Jan 12 2022

From David Wasserman, Nov 14 2006: (Start)
a(n-1) = Sum_{i|d, i < n} a(i-1).
a(p^k-1) = 2^(k-1).
a(n-1) = A067824(n)/2 for n > 1.
a(A122408(n)-1) = A122408(n)/2. (End)
a(A025487(n)-1) = A050324(n). - R. J. Mathar, May 26 2017
a(n) = (A253249(n+1)+1)/4, n > 0. - Geoffrey Critzer, Aug 19 2020

Edited by M. F. Hasler, Oct 12 2018

A074206 Kalmár's [Kalmar's] problem: number of ordered factorizations of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 29 2003

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 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
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 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
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 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
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
If n is squarefree, then a(n) = A000670(A001221(n)) = A000670(A001222(n)). - Vladimir Shevelev and Franklin T. Adams-Watters, Aug 05 2011
A034776 lists the values taken by this sequence. - Robert G. Wilson v, Jun 02 2012
From Gus Wiseman, Aug 25 2020: (Start)
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:
  1  2/1  4/1    6/1    8/1      12/1      30/1
          4/2/1  6/2/1  8/2/1    12/2/1    30/2/1
                 6/3/1  8/4/1    12/3/1    30/3/1
                        8/4/2/1  12/4/1    30/5/1
                                 12/6/1    30/6/1
                                 12/4/2/1  30/10/1
                                 12/6/2/1  30/15/1
                                 12/6/3/1  30/6/2/1
                                           30/6/3/1
                                           30/10/2/1
                                           30/10/5/1
                                           30/15/3/1
                                           30/15/5/1
(End)
a(n) is also the number of ways to tile a strip of length log(n) with tiles having lengths {log(k) : k>=2}. - David Bevan, Jan 07 2025

			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.

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), 1-15.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 10000 terms from T. D. Noe)
David Bevan and Julien Condé, Introducing irrational enumeration: analytic combinatorics for objects of irrational size, arXiv:2412.14682 [math.CO], 2024. See p. 11.
Peter Brown, Number of Ordered Factorizations.
Peter Brown, Number of Ordered Factorizations.
Benny Chor, Paul Lemke, and Ziv Mador, On the number of ordered factorizations of natural numbers, Discrete Math. 214 (2000), no. 1-3, 123-133. MR1743631 (2000m:11093).
Kristin DeVleming and Nikita Singh, Rational unicuspidal plane curves of low degree, arXiv:2311.15922 [math.AG], 2023. See p. 14.
T. M. A. Fink, Properties of the recursive divisor function and the number of ordered factorizations, arXiv:2307.09140 [math.NT], 2023.
E. Hille, A problem in factorisatio numerorum, Acta Arith., 2 (1936), 134-144.
E. Hille, The inversion problem of Möbius, Duke Math. J., 3 (1937), 549-568.
Shikao Ikehara, On Kalmar's Problem in "Factorisatio Numerorum", Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 21 (1939) pp. 208-219.
Shikao Ikehara, On Kalmar's Problem in "Factorisatio Numerorum" II, Proceedings of the Physico-Mathematical Society of Japan. 3rd Series, Vol. 23 (1941) pp. 767-774.
Laszlo Kalmár, Über die mittlere Anzahl der Produktdarstellungen der Zahlen. (Erste Mitteilung), Acta Litt. ac Scient. Szeged 5 (1931): 95-107.
M. Klazar and F. Luca, On the maximal order of numbers in the "factorisatio numerorum" problem, arXiv:math/0505352 [math.NT], 2005-2006.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorization of Integers, The Mathematica Journal, Volume 10, Issue 1 p. 72.
Arnau Mir, Francesc Rossello, and Lucia Rotger, Sound Colless-like balance indices for multifurcating trees, arXiv:1805.01329 [q-bio.PE], 2018.
Augustine O. Munagi, Labeled factorization of integers, INTEGERS: The Electronic Journal of Combinatorics 16:1 (2009), #R50.
L. A. Newberg and D. Naor, A lower bound on the number of solutions to the probed partial digest problem, Advances in Applied Mathematics, 14(2), 1993, 172-183. 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

Apart from initial term, same as A002033.
a(A002110) = A000670, row sums of A251683.
A173382 (and A025523) gives partial sums.
Cf. A008683, A050324, A027751, A001221, A034776, A000670, A050369.
A124433 has these as unsigned row sums.
A334996 has these as row sums.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A008480 counts ordered prime factorizations.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A253249 counts strict chains of divisors.
Cf. A000005, A025487, A046523, A059576, A122408, A124010, A153881 (Dirichlet inverse), A167865, A174725, A174726, A175522, A181819, A320390, A334997, A337105, A361665.

a074206 n | n <= 1 = n
| otherwise = 1 + (sum $ map (a074206 . (div n)) $
tail $ a027751_row n)
-- Reinhard Zumkeller, Oct 01 2012

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 Sellers, Dec 07 2000
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

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 *)
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 *)

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

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

A074206(n)=if(n>1, sumdiv(n, i, if(iA074206(i))),n) \\ M. F. Hasler, Oct 12 2018

A74206=[1]; A074206(n)={if(#A74206A074206(i)))} \\ Use memoization for computing many values. - M. F. Hasler, Oct 12 2018

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

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

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

from math import prod
from functools import lru_cache
from sympy import divisors, factorint, prime
@lru_cache(maxsize=None)
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

@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

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) if k>0 and p is a prime.
Dirichlet g.f.: 1/(2-zeta(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..n-1} a(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 11 2017
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
a(n) = A000670(A001221(n)) for squarefree n. In particular a(A002110(n)) = A000670(n). - Amiram Eldar, May 13 2019
a(n) = A050369(n)/n, for n>=1. - Ridouane Oudra, Aug 31 2019
a(n) = A361665(A181819(n)). - Pontus von Brömssen, Mar 25 2023
From Ridouane Oudra, Nov 02 2023: (Start)
If p,q are distinct primes, and n,m>0 then we have:
a(p^n*q^m) = Sum_{k=0..min(n,m)} 2^(n+m-k-1)*binomial(n,k)*binomial(m,k);
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:
a(n) = Sum_{j=1..bigomega(n)} Sum_{k=1..j} (-1)^(j-k)*binomial(j,k)*tau[k](n), or
a(n) = Sum_{j=1..bigomega(n)} Sum_{k=0..j-1} (-1)^k*binomial(j,k)*tau[j-k](n). (End)

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

A167865 Number of partitions of n into distinct parts greater than 1, with each part divisible by the next.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 3, 3, 3, 1, 5, 1, 5, 4, 3, 1, 6, 2, 5, 4, 5, 1, 9, 1, 6, 4, 4, 4, 8, 1, 6, 6, 7, 1, 11, 1, 8, 8, 4, 1, 10, 3, 10, 5, 8, 1, 11, 4, 10, 7, 6, 1, 13, 1, 10, 11, 7, 6, 15, 1, 9, 5, 11, 1, 14, 1, 9, 12, 8, 5, 15, 1, 16, 9, 8, 1, 18, 5, 12, 7, 10, 1, 21, 7, 13, 11, 5

Offset: 0

Max Alekseyev, Nov 13 2009

Number of lone-child-avoiding achiral rooted trees with n + 1 vertices, where a rooted tree is lone-child-avoiding if all terminal subtrees have at least two branches, and achiral if all branches directly under any given vertex are equal. The Matula-Goebel numbers of these trees are given by A331967. - Gus Wiseman, Feb 07 2020

			a(12) = 4: [12], [10,2], [9,3], [8,4].
a(14) = 3: [14], [12,2], [8,4,2].
a(18) = 5: [18], [16,2], [15,3], [12,6], [12,4,2].
From _Gus Wiseman_, Jul 13 2018: (Start)
The a(36) = 8 lone-child-avoiding achiral rooted trees with 37 vertices:
  (oooooooooooooooooooooooooooooooooooo)
  ((oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo)(oo))
  ((ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo)(ooo))
  ((ooooo)(ooooo)(ooooo)(ooooo)(ooooo)(ooooo))
  ((oooooooo)(oooooooo)(oooooooo)(oooooooo))
  (((ooo)(ooo))((ooo)(ooo))((ooo)(ooo))((ooo)(ooo)))
  ((ooooooooooo)(ooooooooooo)(ooooooooooo))
  ((ooooooooooooooooo)(ooooooooooooooooo))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A001678, A067824, A122651, A167439, A167865, A167866, A184998, A316782.
The semi-achiral version is A320268.
Matula-Goebel numbers of these trees are A331967.
The semi-lone-child-avoiding version is A331991.
Achiral rooted trees are counted by A003238.
Cf. A000081, A000669, A289079, A320222, A331912, A331933, A331936, A331992.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(a((n-d)/d), d=divisors(n) minus{1}))
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Mar 28 2011

a[0] = 1; a[n_] := a[n] = DivisorSum[n, a[(n-#)/#]&, #>1&]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 07 2015 *)

{ A167865(n) = if(n==0,return(1)); sumdiv(n,d, if(d>1, A167865((n-d)\d) ) ) }

a(0) = 1 and for n>=1, a(n) = Sum_{d|n, d>1} a((n-d)/d).

G.f. A(x) satisfies: A(x) = 1 + x^2*A(x^2) + x^3*A(x^3) + x^4*A(x^4) + ... - Ilya Gutkovskiy, May 09 2019

A251683 Irregular triangular array: T(n,k) is the number of ordered factorizations of n with exactly k factors, n >= 1, 1 <= k <= A086436(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 4, 3, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 6, 9, 4, 1, 1, 1, 2, 1, 2, 1, 1, 4, 3, 1, 1, 6, 6, 1, 1, 4, 6, 4, 1, 1, 2, 1, 2, 1, 2, 1, 7, 12, 6, 1, 1, 2, 1, 2, 1, 6, 9, 4

Offset: 1

Geoffrey Critzer, Dec 06 2014

Row sums = A074206.
Row lengths give A086436.
T(n,2) = A070824(n).
T(n,3) = A200221(n).
Sum_{k>=1} k*T(n,k) = A254577.
For all n > 1,  Sum_{k=1..A086436(n)} (-1)^k*T(n,k) = A008683(n). - Geoffrey Critzer, May 25 2018
From Gus Wiseman, Aug 21 2020: (Start)
Also the number of strict length k + 1 chains of divisors from n to 1. For example, row n = 24 counts the following chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1
(End)

			Triangle T(n,k) begins:
  1;
  1;
  1;
  1, 1;
  1;
  1, 2;
  1;
  1, 2, 1;
  1, 1;
  1, 2;
  1;
  1, 4, 3;
  1;
  1, 2;
  1, 2;
  ...
There are 8 ordered factorizations of the integer 12: 12, 6*2, 4*3, 3*4, 2*6, 3*2*2, 2*3*2, 2*2*3.  So T(12,1)=1, T(12,2)=4, and T(12,3)=3.

Alois P. Heinz, Rows n = 1..4000, flattened
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).
Eric Weisstein's World of Mathematics, Ordered Factorization

Cf. A008683, A070824, A200221, A254577.
A008480 gives rows ends.
A086436 gives row lengths.
A124433 is the same except for signs and zeros.
A334996 is the same except for zeros.
A337107 is the restriction to factorial numbers (but with zeros).
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A074206 counts strict chains of divisors from n to 1.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict nonempty chains of divisors of n.
A337071 counts strict chains of divisors starting with n!.
A337256 counts strict chains of divisors of n.
Cf. A001221, A002033, A124010, A167865, A337070, A337105.

with(numtheory):
b:= proc(n) option remember; expand(x*(1+
      add(b(n/d), d=divisors(n) minus {1, n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=1..100);  # Alois P. Heinz, Dec 07 2014

f[1] = {{}};
f[n_] := f[n] =
  Level[Table[
    Map[Prepend[#, d] &, f[n/d]], {d, Rest[Divisors[n]]}], {2}];
Prepend[Map[Select[#, # > 0 &] &,
  Drop[Transpose[
    Table[Map[Count[#, k] &,
      Map[Length, Table[f[n], {n, 1, 40}], {2}]], {k, 1, 10}]],
   1]],{1}] // Grid
(* Second program: *)
b[n_] := b[n] = x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]);
T[n_] := CoefficientList[b[n]/x, x];
Array[T, 100] // Flatten (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Dirichlet g.f.: 1/(1 - y*(zeta(x)-1)).

A334996 Irregular triangle read by rows: T(n, m) is the number of ways to distribute Omega(n) objects into precisely m distinct boxes, with no box empty (Omega(n) >= m).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 4, 3, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 3, 1, 0, 1, 0, 1, 4, 3, 0, 1, 0, 1, 4, 3, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 6, 9, 4, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 3, 0, 1, 0, 1, 6, 6

Offset: 1

Stefano Spezia, May 19 2020

n is the specification number for a set of Omega(n) objects (see Theorem 3 in Beekman's article).
The specification number of a multiset is also called its Heinz number. - Gus Wiseman, Aug 25 2020
From Gus Wiseman, Aug 25 2020: (Start)
For n > 1, T(n,k) is also the number of ordered factorizations of n into k factors > 1. For example, row n = 24 counts the following ordered factorizations (the first column is empty):
  24  3*8   2*2*6  2*2*2*3
      4*6   2*3*4  2*2*3*2
      6*4   2*4*3  2*3*2*2
      8*3   2*6*2  3*2*2*2
      12*2  3*2*4
      2*12  3*4*2
            4*2*3
            4*3*2
            6*2*2
For n > 1, T(n,k) is also the number of strict length-k chains of divisors from n to 1. For example, row n = 36 counts the following chains (the first column is empty):
  36/1  36/2/1   36/4/2/1   36/12/4/2/1
        36/3/1   36/6/2/1   36/12/6/2/1
        36/4/1   36/6/3/1   36/12/6/3/1
        36/6/1   36/9/3/1   36/18/6/2/1
        36/9/1   36/12/2/1  36/18/6/3/1
        36/12/1  36/12/3/1  36/18/9/3/1
        36/18/1  36/12/4/1
                 36/12/6/1
                 36/18/2/1
                 36/18/3/1
                 36/18/6/1
                 36/18/9/1
(End)

			The triangle T(n, m) begins
  n\m| 0     1     2     3     4
  ---+--------------------------
   1 | 0
   2 | 0     1
   3 | 0     1
   4 | 0     1     1
   5 | 0     1
   6 | 0     1     2
   7 | 0     1
   8 | 0     1     2     1
   9 | 0     1     1
  10 | 0     1     2
  11 | 0     1
  12 | 0     1     4     3
  13 | 0     1
  14 | 0     1     2
  15 | 0     1     2
  16 | 0     1     3     3     1
  ...
From _Gus Wiseman_, Aug 25 2020: (Start)
Row n = 36 counts the following distributions of {1,1,2,2} (the first column is empty):
  {1122}  {1}{122}  {1}{1}{22}  {1}{1}{2}{2}
          {11}{22}  {1}{12}{2}  {1}{2}{1}{2}
          {112}{2}  {11}{2}{2}  {1}{2}{2}{1}
          {12}{12}  {1}{2}{12}  {2}{1}{1}{2}
          {122}{1}  {12}{1}{2}  {2}{1}{2}{1}
          {2}{112}  {1}{22}{1}  {2}{2}{1}{1}
          {22}{11}  {12}{2}{1}
                    {2}{1}{12}
                    {2}{11}{2}
                    {2}{12}{1}
                    {2}{2}{11}
                    {22}{1}{1}
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 3000 rows of the table, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

Cf. A000007 (1st column), A000012 (2nd column), A001222 (Omega function), A002033 (row sums shifted left), A007318.
A008480 gives rows ends.
A073093 gives row lengths.
A074206 gives row sums.
A112798 constructs the multiset with each specification number.
A124433 is a signed version.
A251683 is the version with zeros removed.
A334997 is the non-strict version.
A337107 is the restriction to factorial numbers.
A001055 counts factorizations.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A337105 counts strict chains of divisors from n! to 1.
Cf. A007425, A008683, A056239, A124010, A167865, A317144, A319001.

tau[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; (* A334997 *)
T[n_,m_]:=Sum[(-1)^k*Binomial[m,k]*tau[n,m-k-1],{k,0,m-1}]; Table[T[n,m],{n,1,30},{m,0,PrimeOmega[n]}]//Flatten
(* second program *)
chc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]]; (* change {{}} to {} if a(1) = 0 *)
Table[Length[Select[chc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* Gus Wiseman, Aug 25 2020 *)

TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334997
T(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));
tabf(nn) = {for (n=1, nn, print(vector(bigomega(n)+1, k, T(n, k-1))););} \\ Michel Marcus, May 20 2020

T(n, m) = Sum_{k=0..m-1} (-1)^k*binomial(m,k)*tau_{m-k-1}(n), where tau_s(r) = A334997(r, s) (see Theorem 3, Lemma 1 and Lemma 2 in Beekman's article).
Conjecture: Sum_{m=0..Omega(n)} T(n, m) = A002033(n-1) for n > 1.
The above conjecture is true since T(n, m) is also the number of ordered factorizations of n into m factors (see Comments) and A002033(n-1) is the number of ordered factorizations of n. - Stefano Spezia, Aug 21 2025

A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 2, 6, 4, 5, 1, 1, 4, 3, 10, 5, 6, 1, 1, 2, 9, 4, 15, 6, 7, 1, 1, 4, 3, 16, 5, 21, 7, 8, 1, 1, 3, 10, 4, 25, 6, 28, 8, 9, 1, 1, 4, 6, 20, 5, 36, 7, 36, 9, 10, 1, 1, 2, 9, 10, 35, 6, 49, 8, 45, 10, 11, 1, 1, 6, 3, 16, 15, 56, 7, 64, 9, 55, 11, 12, 1

Offset: 1

Stefano Spezia, May 19 2020

T(n, k) is called the generalized divisor function (see Beekman).

As an array with offset n=1, k=0, T(n,k) is the number of length-k chains of divisors of n. For example, the T(4,3) = 10 chains are: 111, 211, 221, 222, 411, 421, 422, 441, 442, 444. - Gus Wiseman, Aug 04 2022

			From _Gus Wiseman_, Aug 04 2022: (Start)
Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4   9  16  25  36  49  64  81
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
The T(4,5) = 21 chains:
  (1,1,1,1,1)  (4,2,1,1,1)  (4,4,2,2,2)
  (2,1,1,1,1)  (4,2,2,1,1)  (4,4,4,1,1)
  (2,2,1,1,1)  (4,2,2,2,1)  (4,4,4,2,1)
  (2,2,2,1,1)  (4,2,2,2,2)  (4,4,4,2,2)
  (2,2,2,2,1)  (4,4,1,1,1)  (4,4,4,4,1)
  (2,2,2,2,2)  (4,4,2,1,1)  (4,4,4,4,2)
  (4,1,1,1,1)  (4,4,2,2,1)  (4,4,4,4,4)
The T(6,3) = 16 chains:
  (1,1,1)  (3,1,1)  (6,2,1)  (6,6,1)
  (2,1,1)  (3,3,1)  (6,2,2)  (6,6,2)
  (2,2,1)  (3,3,3)  (6,3,1)  (6,6,3)
  (2,2,2)  (6,1,1)  (6,3,3)  (6,6,6)
The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
  1
  1  1
  1  2  1
  1  2  3  1
  1  3  3  4  1
  1  2  6  4  5  1
  1  4  3 10  5  6  1
  1  2  9  4 15  6  7  1
  1  4  3 16  5 21  7  8  1
  1  3 10  4 25  6 28  8  9  1
  1  4  6 20  5 36  7 36  9 10  1
  1  2  9 10 35  6 49  8 45 10 11  1
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 150 antidiagonals of the array, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

Cf. A000217 (4th row), A000290 (6th row), A000292 (8th row), A000332 (16th row), A000389 (32nd row), A000537 (36th row), A000578 (30th row), A002411 (12th row), A002417 (24th row), A007318, A027800 (48th row), A335078, A335079.
Column k = 2 of the array is A007425.
Column k = 3 of the array is A007426.
Column k = 4 of the array is A061200.
The transpose of the array is A077592.
The subdiagonal n = k + 1 of the array is A163767.
The version counting all multisets of divisors (not just chains) is A343658.
The strict case is A343662 (row sums: A337256).
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291 counts divisors by Omega.
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A000005, A018892, A051026, A062319, A066959, A176029, A327527, A343652, A343656.

T[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; Table[T[n-k,k],{n,1,13},{k,0,n-1}]//Flatten

T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ Michel Marcus, May 20 2020

T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
T(i*j, k) = T(i, k)*T(j, k) if i and j are coprime positive integers (see Lemma 1 in Beekman's article).
T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).

Duplicate term removed by Stefano Spezia, Jun 03 2020

cp's OEIS Frontend

A122408 Numbers n such that A067824(n) = n.

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A378648 Dirichlet convolution of A067824 and A103977, where A067824 is the number of strict chains of divisors starting with n, and A103977 is the Zumkeller deficiency of n.

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A333952 Recursively highly composite numbers: numbers m such that A067824(m) > A067824(k) for all k < m.

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A378225 Dirichlet inverse of A067824.

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A002033 Number of perfect partitions of n.

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A074206 Kalmár's [Kalmar's] problem: number of ordered factorizations of n.

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