A336426 -id:A336426 - cp's OEIS frontend

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

1, 2, 12, 360, 75600, 174636000, 5244319080000, 2677277333530800000, 25968760179275365452000000, 5793445238736255798985527240000000, 37481813439427687898244906452608585200000000, 7517370874372838151564668004911177464757864076000000000, 55784440720968513813368002533861454979548176771615744085560000000000

Offset: 0

N. J. A. Sloane

Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).
Superprimorials, from primorials by analogy with superfactorials.
Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.
Subsequence of A130091. - Reinhard Zumkeller, May 06 2007
Hankel transform of A171448. - Paul Barry, Dec 09 2009
This might be a good place to explain the name "Chernoff sequence" since his name does not appear in the References or Links as of Mar 22 2014. - Jonathan Sondow, Mar 22 2014
Pickover (1992) named this sequence after Paul Chernoff of California, who contributed this sequence to his book. He was possibly referring to American mathematician Paul Robert Chernoff (1942 - 2017), a professor at the University of California. - Amiram Eldar, Jul 27 2020

			a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.

T. D. Noe, Table of n, a(n) for n=0..25

Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).
Cf. A002110, A051357.
A000142 counts divisors of superprimorials.
A000325 counts uniform divisors of superprimorials.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 is a sister-sequence.
A118914 has row a(n) equal to {1..n}.
A124010 has row a(n) equal to {n..1}.
A130091 lists numbers with distinct prime multiplicities.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
A336426 gives non-products of superprimorials.
Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.

a006939 n = a006939_list !! n
a006939_list = scanl1 (*) a002110_list -- Reinhard Zumkeller, Jul 21 2012

[1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // G. C. Greubel, Oct 14 2018

a := []; printlevel := -1; for k from 0 to 20 do a := [op(a),product(ithprime(i)^(k-i+1),i=1..k)] od; print(a);

Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* Harvey P. Dale, Jul 07 2011 *)
Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* Alonso del Arte, Sep 30 2011 *)

a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ Charles R Greathouse IV, Jul 25 2011

from math import prod
from sympy import prime
def A006939(n): return prod(prime(k)**(n-k+1) for k in range(1,n+1)) # Chai Wah Wu, Aug 12 2025

a(n) = m(1)*m(2)*m(3)*...*m(n), where m(n) = n-th primorial number. - N. J. A. Sloane, Feb 20 2005
a(0) = 1, a(n) = a(n - 1)p(n)#, where p(n)# is the n-th primorial A002110(n) (the product of the first n primes). - Alonso del Arte, Sep 30 2011
log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - Charles R Greathouse IV, Mar 14 2011
A181796(a(n)) = A000110(n+1). It would be interesting to have a bijective proof of this theorem, which is stated at A181796 without proof. See also A336420. - Gus Wiseman, Aug 03 2020

Corrected and extended by Labos Elemer, May 30 2001

A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).

Original entry on oeis.org

1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000

Offset: 0

Clark Kimberling

Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001
a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009
Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

			From _Gus Wiseman_, Aug 12 2020: (Start)
The a(3) = 60 permutations of the prime indices of A006939(3) = 360:
  (111223)  (121123)  (131122)  (212113)  (231211)
  (111232)  (121132)  (131212)  (212131)  (232111)
  (111322)  (121213)  (131221)  (212311)  (311122)
  (112123)  (121231)  (132112)  (213112)  (311212)
  (112132)  (121312)  (132121)  (213121)  (311221)
  (112213)  (121321)  (132211)  (213211)  (312112)
  (112231)  (122113)  (211123)  (221113)  (312121)
  (112312)  (122131)  (211132)  (221131)  (312211)
  (112321)  (122311)  (211213)  (221311)  (321112)
  (113122)  (123112)  (211231)  (223111)  (321121)
  (113212)  (123121)  (211312)  (231112)  (321211)
  (113221)  (123211)  (211321)  (231121)  (322111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..35
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Multinomial Coefficient

Cf. A000178, A014068, A022919, A052295, A208437, A327803.
A190945 counts the case of anti-run permutations.
A317829 counts partitions of this multiset.
A325617 is the version for factorials instead of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008480 counts permutations of prime indices.
A181818 gives products of superprimorials, with complement A336426.
Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.

with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
seq(a(n), n=0..12);  # Alois P. Heinz, May 18 2013

Table[Apply[Multinomial ,Range[n]], {n, 0, 20}]  (* Geoffrey Critzer, Dec 09 2012 *)
Table[Multinomial @@ Range[n], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* Gus Wiseman, Aug 12 2020 *)

a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019

a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.
a(n) = A052295(n)/A000178(n). - Lekraj Beedassy, Feb 19 2004
a(n) = A208437(n*(n+1)/2,n). - Alois P. Heinz, Apr 08 2016
a(n) ~ A * exp(n^2/4 + n + 1/6) * n^(n^2/2 + 7/12) / (2^((n+1)^2/2) * Pi^(n/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 02 2019
a(n) = A327803(n*(n+1)/2,n). - Alois P. Heinz, Sep 25 2019
a(n) = A008480(A006939(n)). - Gus Wiseman, Aug 12 2020

More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
More terms from Michel ten Voorde, Apr 12 2001
Better definition from L. Edson Jeffery, May 18 2013

A181818 Products of superprimorials (A006939).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 360, 384, 512, 576, 720, 768, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 11520, 12288, 13824, 16384, 17280, 18432, 20736, 23040, 24576, 27648, 32768

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

Sorted list of positive integers with a factorization Product p(i)^e(i) such that (e(1) - e(2)) >= (e(2) - e(3)) >= ... >= (e(k-1) - e(k)) >= e(k), with k = A001221(n), and p(k) = A006530(n) = A000040(k), i.e., the prime factors p(1) .. p(k) must be consecutive primes from 2 onward. - Comment clarified by Antti Karttunen, Apr 28 2022
Subsequence of A025487. A025487(n) belongs to this sequence iff A181815(n) is a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A182863. - Matthew Vandermast, May 20 2012

			2, 12, and 360 are all superprimorials (i.e., members of A006939). Therefore, 2*2*12*360 = 17280 is included in the sequence.
From _Gus Wiseman_, Aug 12 2020 (Start):
The sequence of factorizations (which are unique) begins:
    1 = empty product
    2 = 2
    4 = 2*2
    8 = 2*2*2
   12 = 12
   16 = 2*2*2*2
   24 = 2*12
   32 = 2*2*2*2*2
   48 = 2*2*12
   64 = 2*2*2*2*2*2
   96 = 2*2*2*12
  128 = 2*2*2*2*2*2*2
  144 = 12*12
  192 = 2*2*2*2*12
  256 = 2*2*2*2*2*2*2*2
(End)

Amiram Eldar, Table of n, a(n) for n = 1..10000

A181817 rearranged in numerical order. Also includes all members of A000079, A001021, A006939, A009968, A009992, A066120, A166475, A167448, A181813, A181814, A181816, A182763.
Subsequence of A025487, A055932, A087980, A130091, A181824.
A001013 is the version for factorials.
A336426 is the complement.
A336496 is the version for superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A317829 counts factorizations of superprimorials.
Cf. A022915, A076954, A304686, A325368, A336419, A336420, A336421, A353518 (characteristic function).

Select[Range[100],PrimePi[First/@If[#==1,{}, FactorInteger[#]]]==Range[ PrimeNu[#]]&&LessEqual@@Differences[ Append[Last/@FactorInteger[#],0]]&] (* Gus Wiseman, Aug 12 2020 *)

firstdiffs0forward(vec) = { my(v=vector(#vec)); for(n=1,#v,v[n] = vec[n]-if(#v==n,0,vec[1+n])); (v); };
A353518(n) = if(1==n,1,my(f=factor(n), len=#f~); if(primepi(f[len,1])!=len, return(0), my(diffs=firstdiffs0forward(f[,2])); for(i=1,#diffs-1,if(diffs[i+1]>diffs[i],return(0))); (1)));
isA181818(n) = A353518(n); \\ Antti Karttunen, Apr 28 2022

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

Antti Karttunen, Aug 10 2018

nonn

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

			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.

Index entries for sequences related to partitions

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.

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

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 *)

\\ See A318284 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

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

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

A336420 Irregular triangle read by rows where T(n,k) is the number of divisors of the n-th superprimorial A006939(n) with distinct prime multiplicities and k prime factors counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 5, 2, 1, 1, 1, 4, 3, 11, 7, 7, 10, 5, 2, 1, 1, 1, 5, 4, 19, 14, 18, 37, 25, 23, 15, 23, 10, 5, 2, 1, 1, 1, 6, 5, 29, 23, 33, 87, 70, 78, 74, 129, 84, 81, 49, 39, 47, 23, 10, 5, 2, 1, 1, 1, 7, 6, 41, 34, 52, 165, 144, 183, 196, 424, 317, 376, 325, 299, 431, 304, 261, 172, 129, 81, 103, 47, 23, 10, 5, 2, 1, 1

Offset: 0

Gus Wiseman, Jul 25 2020

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
T(n,k) is also the number of length-n vectors 0 <= v_i <= i summing to k whose nonzero values are all distinct.

			Triangle begins:
  1
  1  1
  1  2  1  1
  1  3  2  5  2  1  1
  1  4  3 11  7  7 10  5  2  1  1
  1  5  4 19 14 18 37 25 23 15 23 10  5  2  1  1
The divisors counted in row n = 4 are:
  1  2  4     8   16   48   144   432  2160  10800  75600
     3  9    12   24   72   360   720  3024
     5  25   18   40   80   400  1008
     7       20   54  108   504  1200
             27   56  112   540  2800
             28  135  200   600
             45  189  675   756
             50            1350
             63            1400
             75            4725
            175

A000110 gives row sums.
A000124 gives row lengths.
A000142 counts divisors of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008278 is the version counting only distinct prime factors.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime multiplicities.
A146291 counts divisors by bigomega.
A181796 counts divisors with distinct prime multiplicities.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
A336498 counts divisors of factorials by bigomega.
A336499 uses factorials instead superprimorials.
Cf. A000005, A001222, A008278, A027423, A071625, A124010, A327498, A336419, A336421, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
Table[Length[Select[Divisors[chern[n]],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,5},{k,0,n*(n+1)/2}]

A336417 Number of perfect-power divisors of superprimorials A006939.

Original entry on oeis.org

1, 1, 2, 5, 15, 44, 169, 652, 3106, 15286, 89933, 532476, 3698650, 25749335, 204947216, 1636097441, 14693641859, 132055603656, 1319433514898, 13186485900967, 144978145009105, 1594375302986404, 19128405558986057, 229508085926717076, 2983342885319348522

Offset: 0

Gus Wiseman, Jul 24 2020

nonn

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

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

			The a(0) = 1 through a(4) = 15 divisors:
  1  2  12  360  75600
-------------------------
  1  1   1    1      1
         4    4      4
              8      8
              9      9
             36     16
                    25
                    27
                    36
                   100
                   144
                   216
                   225
                   400
                   900
                  3600

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

A000325 is the uniform version.
A076954 can be used instead of A006939.
A336416 gives the same for factorials instead of superprimorials.
A000217 counts prime power divisors of superprimorials.
A000961 gives prime powers.
A001597 gives perfect powers, with complement A007916.
A006939 gives superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A091050 counts perfect power divisors.
A181818 gives products of superprimorials.
A294068 counts factorizations using perfect powers.
A317829 counts factorizations of superprimorials.
Cf. A000005, A027423, A090630, A118914, A124010, A203025, A251753, A327527, A336419, A336420, A336421, A336426.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[chern[n]],perpouQ]],{n,0,5}]

a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ Andrew Howroyd, Aug 30 2020

a(n) = A091050(A006939(n)).

a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - Andrew Howroyd, Aug 30 2020

Terms a(10) and beyond from Andrew Howroyd, Aug 30 2020

A336419 Number of divisors d of the n-th superprimorial A006939(n) with distinct prime exponents such that the quotient A006939(n)/d also has distinct prime exponents.

Original entry on oeis.org

1, 2, 4, 10, 24, 64, 184, 536, 1608, 5104, 16448, 55136, 187136, 658624, 2339648, 8618208, 31884640, 121733120, 468209408, 1849540416, 7342849216

Offset: 0

Gus Wiseman, Jul 25 2020

A number has distinct prime exponents iff its prime signature is strict.

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

			The a(0) = 1 through a(3) = 10 divisors:
  1  2  12  360
-----------------
  1  1   1    1
     2   3    5
         4    8
        12    9
             18
             20
             40
             45
             72
            360

A000110 shifted once to the left dominates this sequence.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
Cf. A000005, A000178, A008278, A071625, A076954, A118914, A124010, A327498, A327527, A336420, A336421, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
Table[Length[Select[Divisors[chern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,6}]

recurse(n,k,b,d)={if(k>n, 1, sum(i=0, k, if((i==0||!bittest(b,i)) && (i==k||!bittest(d,k-i)), self()(n, k+1, bitor(b, 1<Andrew Howroyd, Aug 30 2020

a(10)-a(20) from Andrew Howroyd, Aug 31 2020

A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

Original entry on oeis.org

1, 3, 13, 76, 571, 5309, 59341, 780149

Offset: 0

Gus Wiseman, Jul 25 2020

A number has distinct prime exponents iff its prime signature is strict.

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

			The a(2) = 13 ways:
  12/1/1  12/2/1  12/3/1  12/4/1  12/12/1
          12/2/2  12/3/3  12/4/2  12/12/2
                          12/4/4  12/12/3
                                  12/12/4
                                  12/12/12

A000258 shifted once to the left is dominated by this sequence.
A336422 is the generalization to non-superprimorials.
A000110 counts divisors of superprimorials with distinct prime exponents.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
Cf. A000005, A008278, A027423, A071625, A118914, A124010, A327498, A336417, A336419, A336420, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
strsig[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[chern[n]],d_?strsig:>Count[Divisors[d],e_?strsig]]],{n,0,5}]

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

A336496 Products of superfactorials (A000178).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, 512, 576, 768, 1024, 1152, 1536, 1728, 2048, 2304, 3072, 3456, 4096, 4608, 6144, 6912, 8192, 9216, 12288, 13824, 16384, 18432, 20736, 24576, 27648, 32768, 34560, 36864, 41472, 49152, 55296

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

First differs from A317804 in having 34560, which is the first term with more than two distinct prime factors.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  512: {1,1,1,1,1,1,1,1,1}

A001013 is the version for factorials, with complement A093373.
A181818 is the version for superprimorials, with complement A336426.
A336497 is the complement.
A000178 lists superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A049711 is the minimum prime multiplicity in A000178.
A174605 is the maximum prime multiplicity in A000178.
A303279 counts prime factors of superfactorials.
A317829 counts factorizations of superprimorials.
A322583 counts factorizations into factorials.
A325509 counts factorizations of factorials into factorials.
Cf. A000142, A000720, A007489, A011371, A022559, A022915, A027423, A034878, A034876, A076954, A115627, A294068.

supfac[n_]:=Product[k!,{k,n}];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[1000],facsusing[Rest[Array[supfac,30]],#]!={}&]

cp's OEIS Frontend

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!*1!*2!*...*n!).

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A181818 Products of superprimorials (A006939).

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

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A336420 Irregular triangle read by rows where T(n,k) is the number of divisors of the n-th superprimorial A006939(n) with distinct prime multiplicities and k prime factors counted with multiplicity.

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A336417 Number of perfect-power divisors of superprimorials A006939.

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).