cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

Original entry on oeis.org

1, 1, 4, 52, 2776, 695541, 927908528, 7303437156115, 371421772559819369, 132348505150329265211927, 355539706668772869353964510735, 7698296698535929906799439134946965681, 1428662247641961794158621629098030994429958386, 2405509035205023556420199819453960482395657232596725626
Offset: 0

Views

Author

Antti Karttunen, Aug 10 2018

Keywords

Comments

Number of factorizations of the superprimorial A006939(n) into factors > 1. - Gus Wiseman, Aug 21 2020

Examples

			For n = 2 we have a multiset {1, 2, 2} which can be partitioned as {{1}, {2}, {2}} or {{1, 2}, {2}} or {{1}, {2, 2}} or {{1, 2, 2}}, thus a(2) = 4.

Links

Crossrefs

Cf. A033312, A317826.
Subsequence of A317828.
A000142 counts submultisets of the same multiset.
A022915 counts permutations of the same multiset.
A337069 is the strict case.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A076716 counts factorizations of factorials.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A181818 lists products of superprimorials, with complement A336426.
Cf. A000178, A002110, A022559, A027423, A124010, A303279, A318284, A322583, A336417, A336496.

Programs

  • Maple
    g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 26 2020
  • Mathematica
    chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[chern[n]]],{n,3}] (* Gus Wiseman, Aug 21 2020 *)
  • PARI
    \\ See A318284 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

Formula

a(n) = A317826(A033312(n+1)) = A317826((n+1)!-1) = A001055(A076954(n)).
a(n) = A001055(A006939(n)). - Gus Wiseman, Aug 21 2020
a(n) = A318284(A002110(n)). - Andrew Howroyd, Aug 31 2020

Extensions

a(0)=1 prepended and a(7) added by Alois P. Heinz, Jul 26 2020
a(8)-a(13) from Andrew Howroyd, Aug 31 2020

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

Views

Author

Gus Wiseman, Aug 17 2020

Keywords

Examples

			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

Links

Crossrefs

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.

Programs

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

Formula

a(n) = A337071(n)/2 for n > 1.
a(n) = A074206(n!).

Extensions

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

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

Views

Author

Gus Wiseman, Aug 16 2020

Keywords

Examples

			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

Crossrefs

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.

Programs

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

Formula

a(n) = 2*A337105(n) for n > 1.
a(n) = A067824(n!).

Extensions

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

A157612 Number of factorizations of n! into distinct factors.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 57, 253, 1060, 5285, 28762, 191263, 1052276, 8028450, 56576192, 424900240, 2584010916, 24952953943, 178322999025, 1886474434192, 15307571683248, 143131274598786, 1423606577935925, 17668243239613767, 137205093278725072, 1399239022852163764, 15774656316828338767
Offset: 0

Views

Author

Jaume Oliver Lafont, Mar 03 2009

Keywords

Comments

The number of factorizations of (n+1)! into k distinct factors can be arranged into the following triangle:
2! 1;
3! 1, 1;
4! 1, 3, 1;
5! 1, 7, 7, 1;
...

Examples

			3! = 6 = 2*3.
a(3) = 2 because there are 2 factorizations of 3!.
4! = 24 = 2*12 = 3*8 = 4*6 = 2*3*4.
a(4) = 5 because there are 5 factorizations of 4!.
5! = 120 (1)
5! = 2*60 = 3*40 = 4*30 = 5*24 = 6*20 = 8*15 = 10*12 (7)
5! = 2*3*20 = 2*4*15 = 2*5*12 = 2*6*10 = 3*4*10 = 3*5*8 = 4*5*6 (7)
5! = 2*3*4*5 (1)
a(5) = 16 because there are 16 factorizations of 5!.

Crossrefs

Cf. A076716, A157017, A157229, A318286. See A157836 for continuation of triangle.

Programs

  • Maple
    with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(n!$2):
seq(a(n), n=0..12);  # Alois P. Heinz, May 26 2013
  • Mathematica
    b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d-1]], {d, Divisors[n] ~Complement~ {1, n}}]];
a[n_] := b[n!, n!];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 21 2017, after Alois P. Heinz *)
  • PARI
    \\ See A318286 for count.
a(n)={if(n<=1, 1, count(factor(n!)[,2]))} \\ Andrew Howroyd, Feb 01 2020

Formula

a(n) = A045778(A000142(n)).

Extensions

a(8)-a(12) from Ray Chandler, Mar 07 2009
a(13)-a(17) from Alois P. Heinz, May 26 2013
a(18)-a(19) from Alois P. Heinz, Jan 10 2015
a(20)-a(26) from Andrew Howroyd, Feb 01 2020

A321468 Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 10, 20, 40, 40, 116, 116, 232, 464, 1440, 1440, 4192, 4192, 11640, 23280, 46560, 46560, 157376
Offset: 0

Views

Author

Gus Wiseman, Nov 11 2018

Keywords

Comments

a(n) is the number of factorizations finer than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

Examples

			The a(2) = 1 through a(8) = 10 factorizations:
2  2*3  2*3*4    2*3*4*5    2*3*4*5*6      2*3*4*5*6*7      2*3*4*5*6*7*8
        2*2*2*3  2*2*2*3*5  2*2*2*3*5*6    2*2*2*3*5*6*7    2*2*2*3*5*6*7*8
                            2*2*3*3*4*5    2*2*3*3*4*5*7    2*2*3*3*4*5*7*8
                            2*2*2*2*3*3*5  2*2*2*2*3*3*5*7  2*2*3*4*4*5*6*7
                                                            2*2*2*2*3*3*5*7*8
                                                            2*2*2*2*3*4*5*6*7
                                                            2*2*2*3*3*4*4*5*7
                                                            2*2*2*2*2*2*3*5*6*7
                                                            2*2*2*2*2*3*3*4*5*7
                                                            2*2*2*2*2*2*2*3*3*5*7
For example, 2*2*2*2*2*2*3*5*6*7 = (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2), so (2*2*2*2*2*2*3*5*6*7) is counted under a(8).

Crossrefs

Dominated by A321514.
Cf. A001055, A066723, A076716, A157612, A242422, A265947, A300383, A317144, A317145, A317534, A321467, A321470, A321471, A321472.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@@Tuples[facs/@Range[2,n]]]],{n,10}]

A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

Original entry on oeis.org

1, 1, 2, 0, 28, 0, 768, 0, 0, 0, 42155360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Aug 16 2020

Keywords

Comments

Support appears to be {0, 1, 2, 4, 6, 10}.

Examples

			The a(4) = 28 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/8/2/1   24/12/4/2/1
      24/3   24/4/1   24/8/4/1
      24/4   24/4/2   24/8/4/2
      24/8   24/8/1   24/12/2/1
      24/12  24/8/2   24/12/3/1
             24/8/4   24/12/4/1
             24/12/1  24/12/4/2
             24/12/2
             24/12/3
             24/12/4

Crossrefs

A336867 is the complement of the support.
A336868 is the characteristic function (image under A057427).
A336942 is half the version for superprimorials (n > 1).
A337071 does not require distinct prime multiplicities.
A337104 is the case of chains ending with 1.
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.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336571 counts chains of divisors 1 < d < n using A130091.
Cf. A000110, A001055, A002033, A007489, A022559, A048656, A071626, A098859, A124010, A167865, A325617, A336416, A336424.

Programs

  • Mathematica
    chnsc[n_]:=If[!UnsameQ@@Last/@FactorInteger[n],{},If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]]];
Table[Length[chnsc[n!]],{n,0,6}]

Formula

a(n) = 2*A337104(n) = 2*A336423(n!) for n > 1.

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

Views

Author

Gus Wiseman, Aug 23 2020

Keywords

Comments

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

Examples

			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

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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!]}]

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200
Offset: 1

Views

Author

Gus Wiseman, Nov 11 2018

Keywords

Examples

			The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:
  (2)*(3)*(4)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

Crossrefs

Dominates A321468.
Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Array[Length[facs[#]]&,n,1,Times],{n,30}]

Formula

a(n) = Product_{k = 1..n} A001055(k).

A093802 Number of distinct factorizations of 105*2^n.

Original entry on oeis.org

5, 15, 36, 74, 141, 250, 426, 696, 1106, 1711, 2593, 3852, 5635, 8118, 11548, 16231, 22577, 31092, 42447, 57464, 77213, 103009, 136529, 179830, 235514, 306751, 397506, 512607, 658030, 841020, 1070490, 1357195, 1714274, 2157539, 2706174, 3383187, 4216358
Offset: 0

Views

Author

Alford Arnold, May 19 2004

Keywords

Examples

			105*A000079 is 105, 210, 420, 840, 1680, 3360, ... and there are 15 distinct factorizations of 210 so a(1) = 15.
a(0) = 5: 105*2^0 = 105 = 3*5*7 = 3*35 = 5*21 = 7*15. - _Alois P. Heinz_, May 26 2013

Links

Crossrefs

Similar sequences: 45*A000079 => A002763, [1, 3, 9, 27, 81, 243...]*A000079 => A054225, 1*A002110 => A000110, 2*A002110 => A035098, A000142 => A076716.
Cf. A001055, A126442, A129306, A346823.
Column k=3 of A346426.

Programs

  • Maple
    with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b((105*2^n)$2):
seq(a(n), n=0..50);  # Alois P. Heinz, May 26 2013
  • Mathematica
    b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0,
     Sum[If[d > k, 0, b[n/d, d]], {d, Divisors[n][[2;;-2]]}]];
a[n_] := b[105*2^n, 105*2^n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

Extensions

2 more terms from Alford Arnold, Aug 29 2007
Corrected offset and extended beyond a(7) by Alois P. Heinz, May 26 2013

A103774 Number of ways to write n! as product of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 2, 6, 10, 42, 42, 82, 204, 1196, 1556, 10324, 34668, 104948, 104964, 873540, 1309396, 11855027, 25238220, 91193575, 453628255, 5002616219, 5902762219, 21142729523, 122981607092, 189706055368, 547296181656, 7291700021313, 14330422534833, 202498591157970
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 15 2005

Keywords

Comments

a(n) = A050320(A000142(n)).
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of n!. For example, The a(2) = 1 through a(6) = 10 set multipartitions are:
{1} {12} {1}{1}{12} {1}{1}{123} {1}{1}{12}{123}
{1}{2} {1}{1}{1}{2} {1}{12}{13} {1}{12}{12}{13}
{1}{1}{1}{23} {1}{1}{1}{12}{23}
{1}{1}{2}{13} {1}{1}{1}{2}{123}
{1}{1}{3}{12} {1}{1}{2}{12}{13}
{1}{1}{1}{2}{3} {1}{1}{3}{12}{12}
{1}{1}{1}{1}{2}{23}
{1}{1}{1}{2}{2}{13}
{1}{1}{1}{2}{3}{12}
{1}{1}{1}{1}{2}{2}{3}
(End)

Examples

			n=5, 5! = 1*2*3*4*5 = 120 = 2 * 2 * 2 * 3 * 5: a(5)=#{2*2*2*3*5,2*2*2*15,2*2*6*5,2*2*30,2*2*3*10,2*6*10}=6.

Links

Crossrefs

A103775 is the strict case.
A157612 is the case of superprimorials.
A001055 counts factorizations.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A076716 counts factorizations of factorials.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

Programs

  • Mathematica
    sub[w_, e_] := Block[{v=w}, v[[e]]--; v]; ric[w_, k_] := ric[w, k] = If[Max[w] == 0, 1, Block[{e, s, p = Flatten@ Position[Sign@w, 1]}, s = Select[ Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; a[n_] := ric[ Sort[ Last /@ FactorInteger[n!]], 1]; Array[a, 22] (* Giovanni Resta, Sep 30 2019 *)

Extensions

a(17)-a(18) from Amiram Eldar, Sep 30 2019
a(19)-a(31) from Giovanni Resta, Sep 30 2019
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