A337070 -id:A337070 - cp's OEIS frontend

A067824 a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 16, 2, 6, 6, 16, 2, 16, 2, 16, 6, 6, 2, 40, 4, 6, 8, 16, 2, 26, 2, 32, 6, 6, 6, 52, 2, 6, 6, 40, 2, 26, 2, 16, 16, 6, 2, 96, 4, 16, 6, 16, 2, 40, 6, 40, 6, 6, 2, 88, 2, 6, 16, 64, 6, 26, 2, 16, 6, 26, 2, 152, 2, 6, 16, 16, 6, 26, 2, 96, 16, 6, 2, 88, 6, 6, 6, 40, 2, 88, 6, 16, 6, 6, 6, 224, 2, 16, 16, 52

Offset: 1

Reinhard Zumkeller, Feb 08 2002

nonn

By a result of Karhumaki and Lifshits, this is also the number of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1}.
The number of tiles of a discrete interval of length n (an interval of Z). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007
Bodini and Rivals proved this is the number of tiles of a discrete interval of length n and also is the number (A107067) of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1} (Bodini, Rivals, 2006). This structure of such tiles is also known as Krasner's factorization (Krasner and Ranulac, 1937). The proof also gives an algorithm to recognize if a set is a tile in optimal time and in this case, to compute the smallest interval it can tile (Bodini, Rivals, 2006). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007
Number of lone-child-avoiding rooted achiral (or generalized Bethe) trees with positive integer leaves summing to n, 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 node are equal. For example, the a(6) = 6 trees are 6, (111111), (222), ((11)(11)(11)), (33), ((111)(111)). - Gus Wiseman, Jul 13 2018. Updated Aug 22 2020.
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of strict chains of divisors starting with n. For example, the a(n) chains for n = 1, 2, 4, 6, 8, 12 are:
  1  2    4      6      8        12
     2/1  4/1    6/1    8/1      12/1
          4/2    6/2    8/2      12/2
          4/2/1  6/3    8/4      12/3
                 6/2/1  8/2/1    12/4
                 6/3/1  8/4/1    12/6
                        8/4/2    12/2/1
                        8/4/2/1  12/3/1
                                 12/4/1
                                 12/4/2
                                 12/6/1
                                 12/6/2
                                 12/6/3
                                 12/4/2/1
                                 12/6/2/1
                                 12/6/3/1
(End)
a(n) is the number of chains including n of the divisor lattice of divisors of n, which is to say, a(n) is the number of (d_1,d_2,...,d_k) such that d_1 < d_2 < ... < d_k = n and d_i divides d_{i+1} for 1 <= i <= k-1. Using this definition, the recurrence a(n) = 1 + Sum_{0 < d < n, d|n} a(d) is evident by enumerating the preceding element of n in the chains. If we count instead the chains whose LCM of members is n, then a(1) would be 2 because the empty chain is included, and we would obtain 2*A074206(n). - Jianing Song, Aug 21 2024

			a(12) = 1 + a(6) + a(4) + a(3) + a(2) + a(1)
= 1+(1+a(3)+a(2)+a(1))+(1+a(2)+a(1))+(1+a(1))+(1+a(1))+(1)
= 1+(1+(1+a(1))+(1+a(1))+1)+(1+(1+a(1))+1)+(1+1)+(1+1)+(1)
= 1+(1+(1+1)+(1+1)+1)+(1+(1+1)+1)+(1+1)+(1+1)+(1)
= 1 + 6 + 4 + 2 + 2 + 1 = 16.

Olivier Bodini and Eric Rivals. Tiling an Interval of the Discrete Line. In M. Lewenstein and G. Valiente, editors, Proc. of the 17th Annual Symposium on Combinatorial Pattern Matching (CPM), volume 4009 of Lecture Notes in Computer Science, pages 117-128. Springer Verlag, 2006.
Juhani Karhumaki, Yury Lifshits and Wojciech Rytter, Tiling Periodicity, in Combinatorial Pattern Matching, Lecture Notes in Computer Science, Volume 4580/2007, Springer-Verlag.

Reinhard Zumkeller, Table = of n, a(n) for n = 1..10000
Olivier Bodini and Eric Rivals, Tiling an Interval of the Discrete Line
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See Table 1 p. 8.
T. M. A. Fink, Properties of the recursive divisor function and the number of ordered factorizations, arXiv:2307.09140 [math.NT], 2023.
Michael Greene and Robin Michaels, One Dimensional Tilings, Eureka (Cambridge) 54 (1996), 4-13.
G. Hajos, Sur le problème de factorisation des groupes cycliques, Acta Math. Acad. Sci. Hung., 1:189-195, 1950.
J. Karhumaki and Y. Lifshits, Tiling periodicity.
M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Eric H. Rivals, Tiling
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001678, A003238, A107067, A107748, A167865, A316782.
Cf. A122408 (fixed points).
Inverse Möbius transform of A074206.
A001055 counts factorizations.
A008480 counts maximal chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A253249 counts nonempty chains of divisors.
A337070 counts chains of divisors starting with A006939(n).
A337071 counts chains of divisors starting with n!.
A337256 counts chains of divisors.
Cf. A001221, A001222, A002033, A124010, A337074, A337105, A378223, A378225 (Dirichlet inverse).

a067824 n = 1 + sum (map a067824 [d | d <- [1..n-1], mod n d == 0])
-- Reinhard Zumkeller, Oct 13 2011

a:= proc(n) option remember;
      1+add(a(d), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 17 2021

a[1]=1; a[n_] := a[n] = 1+Sum[If[Mod[n,d]==0, a[d], 0], {d, 1, n-1}]; Array[a,100] (* Jean-François Alcover, Apr 28 2011 *)

A=vector(100);A[1]=1; for(n=2,#A,A[n]=1+sumdiv(n,d,A[d])); A \\ Charles R Greathouse IV, Nov 20 2012

a(n) = 2*A074206(n), n>1. - Vladeta Jovovic, Jul 03 2005
a(p^k) = 2^k for primes p. - Reinhard Zumkeller, Sep 03 2006
a(n) = Sum_{d|n} A002033(d-1) = Sum_{d|n} A074206(d). - Gus Wiseman, Jul 13 2018
Dirichlet g.f.: zeta(s) / (2 - zeta(s)). - Álvar Ibeas, Dec 30 2018
G.f. A(x) satisfies: A(x) = x/(1 - x) + Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, May 18 2019

Entry revised by N. J. A. Sloane, Aug 27 2006

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)).

A337105 Number of strict chains of divisors from n! to 1.

Original entry on oeis.org

1, 1, 1, 3, 20, 132, 1888, 20128, 584000, 17102016, 553895936, 11616690176, 743337949184, 19467186157568, 999551845713920, 66437400489711616, 10253161206302064640, 388089999627661557760, 53727789519052432998400, 2325767421950553303285760, 365546030278816140131041280

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(4) = 20 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

Andrew Howroyd, Table of n, a(n) for n = 0..200

A325617 is the maximal case.
A336941 is the version for superprimorials.
A337104 counts the case with distinct prime multiplicities.
A337071 is the case not necessarily ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A336423 counts chains using A130091, with maximal case A336569.
A336942 counts chains using A130091 from A006939(n) to 1.
Cf. A001013, A001055, A022559, A124010, A167865, A337070, A337074.

b:= proc(n) option remember; 1 +
      add(b(d), d=numtheory[divisors](n) minus {n})
    end:
a:= n-> ceil(b(n!)/2):
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 23 2020

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

a(n) = A337071(n)/2 for n > 1.

a(n) = A074206(n!).

a(19)-a(20) from Alois P. Heinz, Aug 22 2020

A337256 Number of strict chains of divisors of n.

Original entry on oeis.org

2, 4, 4, 8, 4, 12, 4, 16, 8, 12, 4, 32, 4, 12, 12, 32, 4, 32, 4, 32, 12, 12, 4, 80, 8, 12, 16, 32, 4, 52, 4, 64, 12, 12, 12, 104, 4, 12, 12, 80, 4, 52, 4, 32, 32, 12, 4, 192, 8, 32, 12, 32, 4, 80, 12, 80, 12, 12, 4, 176, 4, 12, 32, 128, 12, 52, 4, 32, 12, 52

Offset: 1

Gus Wiseman, Aug 23 2020

nonn

			The a(n) chains for n = 1, 2, 4, 6, 8 (empty chains shown as 0):
  0  0    0      0      0
  1  1    1      1      1
     2    2      2      2
     2/1  4      3      4
          2/1    6      8
          4/1    2/1    2/1
          4/2    3/1    4/1
          4/2/1  6/1    4/2
                 6/2    8/1
                 6/3    8/2
                 6/2/1  8/4
                 6/3/1  4/2/1
                        8/2/1
                        8/4/1
                        8/4/2
                        8/4/2/1

A067824 is the case of chains starting with n (or ending with 1).
A074206 is the case of chains from n to 1.
A253249 is the nonempty case.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A074206 counts chains of divisors from n to 1.
A122651 counts chains of divisors summing to n.
A167865 counts chains of divisors > 1 summing to n.
A334996 appears to count chains of divisors from n to 1 by length.
A337070 counts chains of divisors starting with A006939(n).
A337071 counts chains of divisors starting with n!.
A337255 counts chains of divisors starting with n by length.
Cf. A001221, A002033, A008480, A124010, A251683, A337105, A337107.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Divisors[n],!(Divisible[#1,#2]||Divisible[#2,#1])&]],{n,10}]

a(n) = A253249(n) + 1.

A337071 Number of strict chains of divisors starting with n!.

Original entry on oeis.org

1, 1, 2, 6, 40, 264, 3776, 40256, 1168000, 34204032, 1107791872, 23233380352, 1486675898368, 38934372315136, 1999103691427840, 132874800979423232, 20506322412604129280, 776179999255323115520, 107455579038104865996800, 4651534843901106606571520, 731092060557632280262082560

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

			The a(1) = 1 through a(3) = 6 chains:
  1  2    6
     2/1  6/1
          6/2
          6/3
          6/2/1
          6/3/1
The a(4) = 40 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

A325617 is the maximal case.
A337070 is the version for superprimorials.
A337074 counts the case with distinct prime multiplicities.
A337105 is the case ending with one.
A000005 counts divisors.
A000142 lists factorial numbers.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
Cf. A001013, A001055, A002033, A022559, A124010, A167865, A336941, A336942.

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

a(n) = 2*A337105(n) for n > 1.

a(n) = A067824(n!).

a(19)-a(20) from Alois P. Heinz, Aug 23 2020

A337255 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 5, 7, 3, 1, 1, 1, 3, 2, 1, 3, 2, 1, 4, 6, 4, 1, 1, 1, 1, 5, 7, 3, 1, 1, 1, 5, 7, 3, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 15, 13, 4, 1, 2, 1, 1, 3, 2, 1, 3, 3, 1, 1, 5, 7, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 23 2020

			Sequence of rows begins:
     1: {1}           16: {1,4,6,4,1}
     2: {1,1}         17: {1,1}
     3: {1,1}         18: {1,5,7,3}
     4: {1,2,1}       19: {1,1}
     5: {1,1}         20: {1,5,7,3}
     6: {1,3,2}       21: {1,3,2}
     7: {1,1}         22: {1,3,2}
     8: {1,3,3,1}     23: {1,1}
     9: {1,2,1}       24: {1,7,15,13,4}
    10: {1,3,2}       25: {1,2,1}
    11: {1,1}         26: {1,3,2}
    12: {1,5,7,3}     27: {1,3,3,1}
    13: {1,1}         28: {1,5,7,3}
    14: {1,3,2}       29: {1,1}
    15: {1,3,2}       30: {1,7,12,6}
Row n = 24 counts the following chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

Alois P. Heinz, Rows n = 1..5000, flattened

A008480 gives rows ends.
A067824 gives row sums.
A073093 gives row lengths.
A334996 appears to be the case of chains ending with 1.
A337071 is the sum of row n!.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A122651 counts chains of divisors summing to n.
A167865 counts chains of divisors > 1 summing to n.
A251683 counts chains of divisors from n to 1 by length.
A253249 counts nonempty chains of divisors.
A337070 counts chains of divisors starting with A006939(n).
A337256 counts chains of divisors.
Cf. A001221, A002033, A124010, A124433, A337074, A337105, A337107.

b:= proc(n) option remember; expand(x*(1 +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 23 2020

chss[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chss[d],{d,Most[Divisors[n]]}],{n}];
Table[Length[Select[chss[n],Length[#]==k&]],{n,30},{k,1+PrimeOmega[n]}]

A336941 Number of strict chains of divisors starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 8, 604, 691968, 16359233536, 10083474928244288, 195661337707783118840768, 139988400203593571474134024847360, 4231553868972506381329450624389969130848256, 6090860257621637852755610879241895108657182173073604608, 464479854191019594417264488167571483344961210693790188774166838214656

Offset: 0

Gus Wiseman, Aug 13 2020

nonn

			The a(2) = 8 chains:
  12/1
  12/2/1
  12/3/1
  12/4/1
  12/6/1
  12/4/2/1
  12/6/2/1
  12/6/3/1

Andrew Howroyd, Table of n, a(n) for n = 0..25

A022915 is the maximal case.
A076954 can be used instead of A006939.
A336571 is the case with distinct prime multiplicities.
A336942 is the case using members of A130091.
A337070 is the version ending with any divisor of A006939(n).
A000005 counts divisors.
A074206 counts chains of divisors from n to 1.
A006939 lists superprimorials or Chernoff numbers.
A067824 counts divisor chains starting with n.
A181818 gives products of superprimorials, with complement A336426.
A253249 counts chains of divisors.
A317829 counts factorizations of superprimorials.
A336423 counts chains using A130091, with maximal case A336569.
Cf. A000142, A001055, A002033, A008480, A022559, A027423, A124010, A167865, A181796, A336417, A336420, A337069.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chns[n_]:=If[n==1,1,Sum[chns[d],{d,Most[Divisors[n]]}]];
Table[chns[chern[n]],{n,0,3}]

a(n)={my(sig=vector(n,i,i), 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

a(n) = A337070(n)/2 for n > 0.

a(n) = A074206(A006939(n)).

Terms a(8) and beyond from Andrew Howroyd, Aug 30 2020

A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 9, 4, 0, 1, 14, 45, 52, 20, 0, 1, 28, 183, 496, 655, 420, 105, 0, 1, 58, 633, 2716, 5755, 6450, 3675, 840, 0, 1, 94, 1659, 11996, 46235, 106806, 155869, 145384, 84276, 27720, 3960

Offset: 1

Gus Wiseman, Aug 23 2020

Row n > 1 appears to be row n! of A334996.

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    9    4
    0    1   14   45   52   20
    0    1   28  183  496  655  420  105
    0    1   58  633 2716 5755 6450 3675  840
Row n = 4 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

A097805 is the restriction to powers of 2.
A325617 is the maximal case.
A337105 gives row sums.
A337106 is column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A074206 counts chains of divisors from n to 1.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains starting with n!.
Cf. A022559, A124010, A167865, A251683, A337070, A337074.

b:= proc(n) option remember; expand(x*(`if`(n=1, 1, 0) +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n!)):
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 23 2020

nv=5;
chnsc[n_]:=Select[Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],n]}],{n}],MemberQ[#,1]&];
Table[Length[Select[chnsc[n!],Length[#]==k&]],{n,nv},{k,1+PrimeOmega[n!]}]

A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Gus Wiseman, May 01 2021

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)

Column k = 1 is A000005.
Row ends are A008480.
Row lengths are A073093.
Column k = 2 is A238952.
The case from n to 1 is A334996 or A251683 (row sums: A074206).
A non-strict version is A334997 (transpose: A077592).
The case starting with n is A337255 (row sums: A067824).
Row sums are A337256 (nonempty: A253249).
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]

A337069 Number of strict factorizations of the superprimorial A006939(n).

Original entry on oeis.org

1, 1, 3, 34, 1591, 360144, 442349835, 3255845551937, 156795416820025934, 53452979022001011490033, 138542156296245533221812350867, 2914321438328993304235584538307144802, 528454951438415221505169213611461783474874149, 873544754831735539240447436467067438924478174290477803

Offset: 0

Gus Wiseman, Aug 15 2020

nonn

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

Also the number of strict multiset partitions of {1,2,2,3,3,3,...,n}, a multiset with i copies of i for i = 1..n.

			The a(3) = 34 factorizations:
  2*3*4*15  2*3*60   2*180  360
  2*3*5*12  2*4*45   3*120
  2*3*6*10  2*5*36   4*90
  2*4*5*9   2*6*30   5*72
  3*4*5*6   2*9*20   6*60
            2*10*18  8*45
            2*12*15  9*40
            3*4*30   10*36
            3*5*24   12*30
            3*6*20   15*24
            3*8*15   18*20
            3*10*12
            4*5*18
            4*6*15
            4*9*10
            5*6*12
            5*8*9

A022915 counts permutations of the same multiset.
A157612 is the version for factorials instead of superprimorials.
A317829 is the non-strict version.
A337072 is the non-strict version with squarefree factors.
A337073 is the case with squarefree factors.
A000217 counts prime factors (with multiplicity) of superprimorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A181818 lists products of superprimorials, with complement A336426.
A322583 counts factorizations into factorials.
Cf. A000142, A000178, A002110, A022559, A027423, A303279, A318286, A322583, A337070, A337071.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
stfa[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfa[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Length[stfa[chern[n]]],{n,0,3}]

\\ See A318286 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

a(n) = A045778(A006939(n)).

a(n) = A318286(A002110(n)). - Andrew Howroyd, Sep 01 2020

a(7)-a(13) from Andrew Howroyd, Sep 01 2020

cp's OEIS Frontend

A067824 a(1) = 1; for n > 1, a(n) = 1 + Sum_{0 < d < n, d|n} a(d).

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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).

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A337105 Number of strict chains of divisors from n! to 1.

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A337256 Number of strict chains of divisors of n.

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A337071 Number of strict chains of divisors starting with n!.

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A337255 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.

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A336941 Number of strict chains of divisors starting with the superprimorial A006939(n) and ending with 1.

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