A050324 -id:A050324 - 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.

A051282 2-adic valuation of A025487: largest k such that 2^k divides A025487(n), where A025487 gives products of primorials.

Original entry on oeis.org

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

Offset: 1

Alford Arnold

a(n) can be used for resorting A025487 and sequences indexed by A025487, e.g., A050322, A050323, A050324 and A050325.

a(n) is the number of primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers. - Amiram Eldar, Jun 03 2023

			a(8) = 3 because A025487(8) = 24 and 2^3 divides 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001055, A002033, A007814, A025487, A045778, A050320, A051903.

Cf. A050322, A050323, A050324, A050325.

a051282 = a007814 . a025487  -- Reinhard Zumkeller, Apr 06 2013

max = 40000; A025487 = {1}; lpe = {}; Do[ pe = Sort[ FactorInteger[n][[All, 2]]]; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[A025487, n]], {n, 2, max}]; a[n_] := FactorInteger[ A025487[[n]] ][[1, 2]]; a[1] = 0; Table[a[n], {n, 1, Length[A025487]}] (* Jean-François Alcover, Jun 14 2012, after Robert G. Wilson v *)

isA025487(n)=my(k=valuation(n, 2), t); n>>=k; forprime(p=3, default(primelimit), t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1)))
[valuation(n,2) | n <- [1..1000], isA025487(n)]
\\ Or, for older versions:
apply(n->valuation(n,2), select(isA025487, [1..1000])) \\ Charles R Greathouse IV, Nov 07 2014

a(n) = A007814(A025487(n)) = A051903(A025487(n)). - Matthew Vandermast, Jul 03 2012

More terms from Naohiro Nomoto, Mar 11 2001

A095705 Triangular array read by rows: a(n, k) = sum of number of ordered factorizations of all prime signatures with n total prime factors and k distinct prime factors.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 46, 44, 75, 16, 124, 308, 308, 541, 32, 572, 1790, 2536, 2612, 4683, 64, 1568, 8352, 17028, 24704, 25988, 47293, 128, 6728, 40628, 137498, 187928, 277456, 296564, 545835, 256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261

Offset: 1

Alford Arnold, Jul 04 2004, Nov 22 2005

A093936 is an analogous array for unordered factorizations.

First column is A000079. First two diagonals are A000670 and A005649.

			There are two prime signatures with 5 total primes and 3 distinct primes: p^3*q*r and p^2*q^2*r. A074206(p^3*q*r) = 132 and A074206(p^2*q^2*r) = 176, so a(5, 3) = 132+176 = 308.
Array begins:
  1
  2, 3
  4, 8, 13
  8, 46, 44, 75
  16, 124, 308, 308, 541
  32, 572, 1790, 2536, 2612, 4683
  64, 1568, 8352, 17028, 24704, 25988, 47293
  128, 6728, 40628, 137498, 187928, 277456, 296564, 545835
  256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261

Amiram Eldar, Table of n, a(n) for n = 1..210 (rows n=1..20, flattened)

A035341 gives the row sums. Cf. A050324, A074206, A093936, A096443.

H[0] = 0; H[1] = 1; H[n_] := H[n] = Total[H /@ Most[Divisors[n]]];
T[n_, k_] := Module[{t = IntegerPartitions[n, {k}]}, Total[H /@ Times @@@ ((Prime[Range[k]]^#) & /@ t)]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 28 2025 *)

Edited and extended by David Wasserman, Feb 22 2008

A098348 Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 16, 48, 132, 308, 541, 32, 112, 368, 1076, 2612, 4683, 64, 256, 976, 3408, 10404, 25988, 47293, 128, 576, 2496, 10096, 36848, 116180, 296564, 545835, 256, 1280, 6208, 28480, 120400, 454608, 1469892, 3816548

Offset: 1

Alford Arnold, Sep 04 2004

The first three columns are A000079, A001792 and A098385.
The first two diagonals are A000670 and A005649.
A070175 gives the smallest representative of each hook-type prime signature, so this sequence is a rearrangement of A074206(A070175).

			a(4, 2) = 20 because 24=2*2*2*3 has 20 ordered factorizations and so does any other number with the same prime signature.

Cf. A050324, A070175, A070826, A074206, A095705. A098349 gives the row sums. A098384.

a(n, k) = 1 + (Sum_{i=1..k-1} binomial(k-1, i)*a(i, i)) + (Sum_{j=1..k} Sum_{i=j..j+n-k-1} binomial(k-1, j-1)*a(i, j)) + (Sum_{j=1..k-1} binomial(k-1,j-1)*a(j+n-k, j)). - David Wasserman, Feb 21 2008
a(n, k) = A074206(2^(n+1-k)*A070826(k)). - David Wasserman, Feb 21 2008
The following conjectural formula for the triangle entries agrees with the values listed above: T(n,k) = Sum_{j = 0..n-k} 2^(n-k-j)*binomial(n-k,j)*a(k,j), where a(k,j) = 2^j*Sum_{i = j+1..k+1} binomial(i,j+1)*(i-1)!*Stirling2(k+1,i). See A098384 for related conjectures. - Peter Bala, Apr 20 2012

Edited and extended by David Wasserman, Feb 21 2008

A173382 Partial sums of A074206.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 9, 10, 14, 16, 19, 20, 28, 29, 32, 35, 43, 44, 52, 53, 61, 64, 67, 68, 88, 90, 93, 97, 105, 106, 119, 120, 136, 139, 142, 145, 171, 172, 175, 178, 198, 199, 212, 213, 221, 229, 232, 233, 281, 283, 291, 294, 302, 303, 323, 326, 346, 349, 352, 353, 397, 398, 401, 409, 441, 444, 457

Offset: 0

Jonathan Vos Post, Feb 17 2010

Partial sums of number of ordered factorizations of n.

a(n) is also the number of ways to tile a strip of length at most log(n) with tiles having lengths {log(k) : k>=2}. - David Bevan, Jan 07 2025

			a(96) = 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.

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.
Kalmár, Laszlo. "Über die mittlere Anzahl der Produktdarstellungen der Zahlen.(Erste Mitteilung)'." Acta Litt. ac Scient. Szeged 5 (1931): 95-107.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
David Bevan and Julien Condé, Introducing irrational enumeration: analytic combinatorics for objects of irrational size, arXiv:2412.14682 [math.CO], 2024. See p. 11.
Ann Clifton, Eva Czabarka, Kevin Liu, Sarah Loeb, Utku Okur, Laszlo Szekely, and Kristina Wicke, Universal rooted phylogenetic tree shapes and universal tanglegrams, arXiv:2308.06580 [math.CO], 2023.
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A074206, A002033, A001055, A050324, A000670.

A025523 is an essentially identical sequence.

Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = 1 + Sum[a[Floor[n/k]], {k, 2, n}]; Table[a[n], {n, 0, 100}] (* Vaclav Kotesovec, Jan 31 2019 *)
Clear[a]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; Join[{0}, Accumulate[a /@ Range[100]]] (* Vaclav Kotesovec, Jan 31 2019, after Jean-François Alcover, faster *)

a(n) = Sum_{i=0..n} A074206(i).

a(n) ~ -n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation Zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019

Terms corrected by N. J. A. Sloane, May 04 2016

A334256 Numbers k such that H(k) = 2*k, where H(k) is the number of ordered factorizations of k (A074206).

Original entry on oeis.org

3072, 1310720, 469762048, 48378511622144, 14636698788954112, 1115414963960152064, 1254378597012249509888, 358899852698093036240896, 28472620903563746322679857152

Offset: 1

David A. Corneth and Amiram Eldar, Apr 20 2020

If p is an odd prime then 2^(4*p - 2) * p is a term, hence this sequence is infinite.
Since A074206(k) depends only on the prime signature (A124010) of k, then each term is of the form A050324(k)/2 = A074206(A025487(k))/2.
Besides terms of the form 2^(4*p - 2) * p at least 79 terms not of this form are known. For example, 1115414963960152064 = 2^46 * 11^2 * 131 is a term not of this form. To ease the search, can we narrow the possible prime signatures of terms?

			3072 is a term since A074206(3072) = 6144 = 2 * 3072.

David A. Corneth, List of found terms in form of oddpart * 2^(multiplicity of 2)

Subsequence of A270308.
Cf. A074206, A122408, A163272.
Cf. A025487, A050324, A124010.

h[1] = 1; h[n_] := h[n] = DivisorSum[n, h[#] &, # < n &]; Select[Range[1.5*10^6], h[#] == 2*# &]

PARI
```
is(n) = A074206(n) == n<<1
```

A131420 A tabular sequence of arrays counting ordered factorizations over least prime signatures. The unordered version is described by sequence A129306.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 26, 16, 48, 132, 308, 541, 76, 176, 32, 112, 368, 1076, 2612, 4683, 208, 604, 1460, 252, 818, 64, 256, 876, 3408, 10404, 25988, 47293, 544, 1888, 5740, 14300, 768, 2316, 3172, 7880, 128, 576, 2496, 10096, 36848, 116180

Offset: 1

Alford Arnold, Jul 10 2007

nonn

The display has 1 2 3 5 7 11 15 ... terms per column. (cf. A000041)
The arrays begin
1.....2.....4......8......16.....32.....64......128
......3.....8.....20......48....112....256......576
...........13.....44.....132....368....976.....2496
..................75.....308...1076...3408....10096
.........................541...2612..10404....36848
...............................4683..25988...116180
.....................................47293...296564
.............................................545835
..................26......76....208....544
.........................176....604...1888
...............................1460...5740
.....................................14300
................................252....768
......................................2316
................................818...3172
......................................7880
with column sums
1....5....25....173....1297....12225....124997 => A035341
Column i corresponds to partitions of i. The rows correspond successively to the partitions {i}, {i-1,1},{i-2,1,1},{i-3,1,1,1}, ..., {i-7,1,1,1,1,1,1,1}, {i-2,2}, {i-3,2,1}, {i-4,2,1,1}, {i-5,2,1,1,1}, {i-3,3}, {i-3,3,1}, {i-4,2,2}, {i-5,2,2,1}. - Roger Lipsett, Feb 26 2016

			36 = 2*2*3*3 and is in A025487. There are 26 ways to factor 36 so a(11) = 26.

Cf. A000041, A000670, A002033, A025487, A035098, A035341, A050324, A074206, A095705, A098348, A104725, A108464, A129306.

gozinta counts ordered factorizations of an integer, and if lst is a partition we have
gozinta[1] = 1;
gozinta[n_] := gozinta[n] = 1 + Sum[gozinta[n/i], {i, Rest@Most@Divisors@n}]
a[lst_] := gozinta[Times @@ (Array[Prime, Length@lst]^lst)] (* Roger Lipsett, Feb 26 2016 *)

Corrected entries in table in comments section - Roger Lipsett, Feb 26 2016

cp's OEIS Frontend

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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A051282 2-adic valuation of A025487: largest k such that 2^k divides A025487(n), where A025487 gives products of primorials.

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A095705 Triangular array read by rows: a(n, k) = sum of number of ordered factorizations of all prime signatures with n total prime factors and k distinct prime factors.

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A098348 Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.

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A173382 Partial sums of A074206.

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