A327523 -id:A327523 - cp's OEIS frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

This sequence does not contain any number of the form 36n-6 or 36n+6, as such numbers are divisible by 6 but not by 4 or 9. Consequently, this sequence does not contain 24 consecutive integers. The quest for the greatest number of consecutive integers in this sequence has ties to the ABC conjecture (see the MathOverflow link). - Danny Rorabaugh, Sep 23 2015
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities. The enumeration of these partitions by sum is given by A098859. - Gus Wiseman, May 04 2019
Aktaş and Ram Murty (2017) called these terms "special numbers" ("for lack of a better word"). They prove that the number of terms below x is ~ c*x/log(x), where c > 1 is a constant. - Amiram Eldar, Feb 25 2021
Sequence A005940(1+A328592(n)), n >= 1, sorted into ascending order. - Antti Karttunen, Apr 03 2022

			From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
MathOverflow, Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, arXiv:1902.09224 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Factorization

Complement of A130092. A351564 is the characteristic function.
Subsequence of A351294.
Cf. A000961, A006939, A181818, A304686, A319161, A342028, A342029, A342030, A342031, A342032 (subsequences).
Cf. A005940, A048767, A048768, A056239, A098859, A112798, A118914, A181796, A217605, A325326, A325337, A325368, A327498, A327523, A328592, A336423, A336424, A336569, A336570, A336571, A343012, A343013.

filter:= proc(t) local f;
f:= map2(op,2,ifactors(t)[2]);
nops(f) = nops(convert(f,set));
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 30 2015

t[n_] := FactorInteger[n][[All, 2]]; Select[Range[400],  Union[t[#]] == Sort[t[#]] &]  (* Clark Kimberling, Mar 12 2015 *)

isok(n) = {nbf = omega(n); f = factor(n); for (i = 1, nbf, for (j = i+1, nbf, if (f[i, 2] == f[j, 2], return (0)););); return (1);} \\ Michel Marcus, Aug 18 2013

isA130091(n) = issquarefree(factorback(apply(e->prime(e), (factor(n)[, 2])))); \\ Antti Karttunen, Apr 03 2022

a(n) < A130092(n) for n<=150, a(n) > A130092(n) for n>150.

A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 7, 2, 3, 3, 7, 2, 4, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3, 7, 2, 7, 3, 5, 3, 3, 3, 11, 2, 5, 5, 7, 2, 4, 2, 7, 4

Offset: 1

Matthew Vandermast, Nov 22 2010

nonn

The canonical factorization of n into prime powers can be written as Product p(i)^e(i), for example. A host of equivalent notations can also be used (for another example, see Weisstein link). a(n) depends only on prime signature of n (cf. A025487).
a(n) >= A085082(n). (A085082(n) equals the number of members of A025487 that divide A046523(n), and each member of A025487 is divisible by at least one member of A130091 that divides no smaller member of A025487.) a(n) > A085082(n) iff n has in its canonical prime factorization at least two exponents greater than 1.
a(n) = number of such divisors of n that in their prime factorization all exponents are unique. - Antti Karttunen, May 27 2017
First differs from A335549 at a(90) = 7, A335549(90) = 8. First differs from A335516 at a(180) = 9, A335516(180) = 10. - Gus Wiseman, Jun 28 2020

			12 has a total of six divisors (1, 2, 3, 4, 6 and 12). Of those divisors, the number 1 has no prime factors, hence, no positive exponents at all (and no repeated positive exponents) in its canonical prime factorization. The lists of positive exponents for 2, 3, 4, 6 and 12 are (1), (1), (2), (1,1) and (2,1) respectively (cf. A124010). Of all six divisors, only the number 6 (2^1*3^1) has at least one positive exponent repeated (namely, 1). The other five do not; hence, a(12) = 5.
For n = 90 = 2 * 3^2 * 5, the divisors that satisfy the condition are: 1, 2, 3, 3^2, 5, 2 * 3^2, 3^2 * 5, altogether 7, (but for example 90 itself is not included), thus a(90) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Gus Wiseman, Sequences counting and encoding certain classes of multisets

Diverges from A088873 at n=24 and from A085082 at n=36. a(36) = 7, while A085082(36) = 6.
Partitions with distinct multiplicities are A098859.
Sorted prime signature is A118914.
Unsorted prime signature is A124010.
a(n) is the number of divisors of n in A130091.
Factorizations with distinct multiplicities are A255231.
The largest of the counted divisors is A327498.
Factorizations using the counted divisors are A327523.
Cf. A000005, A007916, A045778, A056239, A212168, A327498, A335519, A335549.

Table[DivisorSum[n, 1 &, Length@ Union@ # == Length@ # &@ FactorInteger[#][[All, -1]] &], {n, 105}] (* Michael De Vlieger, May 28 2017 *)

no_repeated_exponents(n) = { my(es = factor(n)[, 2]); if(length(Set(es)) == length(es),1,0); }
A181796(n) = sumdiv(n,d,no_repeated_exponents(d)); \\ Antti Karttunen, May 27 2017

from sympy import factorint, divisors
def ok(n):
    f=factorint(n)
    ex=[f[i] for i in f]
    for i in ex:
        if ex.count(i)>1: return 0
    return 1
def a(n): return sum([1 for i in divisors(n) if ok(i)]) # Indranil Ghosh, May 27 2017

a(A000079(n)) = a(A002110(n)) = n+1.
a(A006939(n)) = A000110(n+1).
a(A181555(n)) = A002720(n).

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 03 2020

nonn

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 a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
  2  4    8      5*12     16       4*9      32         48
     2*2  2*4    3*20     4*4      3*12     4*8        4*12
          2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
                 2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
                          2*2*2*2  2*2*9    2*2*8      2*24
                                   2*2*3*3  2*2*2*4    2*3*8
                                            2*2*2*2*2  2*2*12
                                                       2*2*3*4
                                                       2*2*2*2*3

A327523 is the case when n is restricted to belong to A130091 also.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty chains of divisors.
A281116 counts factorizations with no common divisor.
A302696 lists numbers whose prime indices are pairwise coprime.
A305149 counts stable factorizations.
A320439 counts factorizations using A289509.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A336568 = not a product of two numbers with distinct prime multiplicities.
A336569 counts maximal chains of elements of A130091.
A337256 counts chains of divisors.
Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.

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,#]&]}]];
Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]

A336571 Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 5, 1, 5, 3, 3, 1, 14, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 3, 17, 1, 3, 3, 14, 1, 4, 1, 5, 5, 3, 1, 36, 2, 5, 3, 5, 1, 14, 3, 14, 3, 3, 1, 16, 1, 3, 5, 32, 3, 4, 1, 5, 3, 4, 1, 35, 1, 3, 5, 5, 3, 4, 1, 36, 8, 3, 1

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

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 a(n) sets for n = 4, 6, 12, 16, 24, 84, 36:
  {}   {}   {}     {}       {}        {}        {}
  {2}  {2}  {2}    {2}      {2}       {2}       {2}
       {3}  {3}    {4}      {3}       {3}       {3}
            {4}    {8}      {4}       {4}       {4}
            {2,4}  {2,4}    {8}       {7}       {9}
                   {2,8}    {12}      {12}      {12}
                   {4,8}    {2,4}     {28}      {18}
                   {2,4,8}  {2,8}     {2,4}     {2,4}
                            {4,8}     {2,12}    {3,9}
                            {2,12}    {2,28}    {2,12}
                            {3,12}    {3,12}    {2,18}
                            {4,12}    {4,12}    {3,12}
                            {2,4,8}   {4,28}    {3,18}
                            {2,4,12}  {7,28}    {4,12}
                                      {2,4,12}  {9,18}
                                      {2,4,28}  {2,4,12}
                                                {3,9,18}

A336423 is the version for chains containing n.
A336570 is the maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
Cf. A001222, A002033, A005117, A098859, A118914, A124010, A294068, A305149, A327498, A327523, A336568, A336569.

strchns[n_]:=If[n==1,1,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[strchns[n],{n,100}]

A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 1, 2, 2, 0, 1, 4, 1, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

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 a(n) chains for n = 12, 72, 144, 192 (ones not shown):
  12/3    72/18/2       144/72/18/2       192/96/48/24/12/3
  12/4/2  72/18/9/3     144/72/18/9/3     192/64/32/16/8/4/2
          72/24/12/3    144/48/24/12/3    192/96/32/16/8/4/2
          72/24/8/4/2   144/72/24/12/3    192/96/48/16/8/4/2
          72/24/12/4/2  144/48/16/8/4/2   192/96/48/24/8/4/2
                        144/48/24/8/4/2   192/96/48/24/12/4/2
                        144/72/24/8/4/2
                        144/48/24/12/4/2
                        144/72/24/12/4/2

A336423 is the non-maximal version.
A336570 is the version for chains not necessarily containing n.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336571 counts divisor sets of elements of A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strchs[n_]:=If[n==1,{{}},If[!strsigQ[n],{},Join@@Table[Prepend[#,d]&/@strchs[d],{d,Select[Most[Divisors[n]],strsigQ]}]]];
Table[Length[fasmax[strchs[n]]],{n,100}]

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 3, 1, 4, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

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 a(n) sets for n = 36, 120, 144, 180 (ones not shown):
  {2,18}    {3,12,24}    {2,18,72}       {2,18}
  {3,12}    {5,20,40}    {3,9,18,72}     {3,12}
  {2,4,12}  {2,4,8,24}   {3,12,24,48}    {5,20}
  {3,9,18}  {2,4,8,40}   {3,12,24,72}    {5,45}
            {2,4,12,24}  {2,4,8,16,48}   {2,4,12}
            {2,4,20,40}  {2,4,8,24,48}   {2,4,20}
                         {2,4,8,24,72}   {3,9,18}
                         {2,4,12,24,48}  {3,9,45}
                         {2,4,12,24,72}

A336569 is the version for chains containing n.
A336571 is the non-maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336423, A336425, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strses[n_]:=If[n==1,{{}},Join@@Table[Append[#,d]&/@strses[d],{d,Select[Most[Divisors[n]],strsigQ]}]];
Table[Length[fasmax[strses[n]]],{n,100}]

A337104 Number of strict chains of divisors from n! to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 0, 14, 0, 384, 0, 0, 0, 21077680, 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

Gus Wiseman, Aug 17 2020

nonn

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

			The a(4) = 14 chains:
  24/1
  24/2/1
  24/3/1
  24/4/1
  24/8/1
  24/12/1
  24/4/2/1
  24/8/2/1
  24/8/4/1
  24/12/2/1
  24/12/3/1
  24/12/4/1
  24/8/4/2/1
  24/12/4/2/1

A336867 appears to be the positions of zeros.
A336868 is the characteristic function (image under A057427).
A336942 is the version for superprimorials (n > 1).
A337105 does not require distinct prime multiplicities.
A337074 does not require chains to end with 1.
A337075 is the version for chains not containing n!.
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.
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.
A336423 counts chains using A130091, with maximal case A336569.
A336425 counts divisible pairs of divisors of n!, both in A130091.
A336571 counts chains of divisors 1 < d < n using A130091.
A337071 counts chains of divisors starting with n!.
Cf. A002033, A022559, A071626, A098859, A124010, A167865, A327523, A336424, A336941.

strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[DeleteCases[Divisors[n],n],UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n!],{n,0,8}]

a(n) = A337075(n) whenever A337075(n) != 0.
a(n) = A337074(n)/2 for n > 1.
a(n) = A336423(n!).

A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

Original entry on oeis.org

1, 1, 1, 3, 14, 48, 384, 1308, 40288, 933848, 21077680, 75690016, 5471262080, 7964665440, 54595767744, 17948164982144, 3454946386353664, 5010658671663616, 723456523262697984, 950502767770273280, 165679731871366906880, 8443707247468681128448

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(1) = 1 through a(4) = 14 chains (with n! prepended):
  1  2/1  6/1    24/1
          6/2/1  24/2/1
          6/3/1  24/3/1
                 24/4/1
                 24/8/1
                 24/12/1
                 24/4/2/1
                 24/8/2/1
                 24/8/4/1
                 24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/8/4/2/1
                 24/12/4/2/1

A336571 is the generalization to not just factorial numbers.
A337104 is the version for chains containing n!.
A000005 counts divisors.
A001055 counts factorizations.
A032741 counts proper divisors.
A071625 counts distinct prime multiplicities.
A074206 counts chains of divisors from n to 1.
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.
A336423 counts chains using A130091, with maximal case A336569.
A336424 counts factorizations using A130091.
A336425 counts divisible pairs of divisors of n!, both in A130091.
Cf. A002033, A098859, A124010, A294068, A327499, A327500, A327523, A337071.

chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[n!],{n,0,5}]

a(n) = A337104(n) whenever A337104(n) != 0.

a(n) = A336571(n!).

A336871 Number of divisors d of A076954(n) with distinct prime multiplicities such that the numerator of A006939(n)/d also has distinct prime multiplicities.

Original entry on oeis.org

1, 2, 4, 11, 28, 96, 309, 1256, 4676, 21647

Offset: 0

Gus Wiseman, Aug 06 2020

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

The sequence A076954 is A076954(n) = Product_{i=1..n} prime(i)^i.

			The a(0) = 1 through a(3) = 11 divisors:
  1  2  18   2250
     1   9   1125
         3    375
         1    125
               75
               45
               25
               18
                9
                5
                1

A336419 is the version for superprimorials.
A336500 is the generalization to all positive integers.
A000005 counts divisors.
A006939 lists superprimorials or Chernoff numbers.
A007425 counts divisors of divisors.
A076954 is a sister of superprimorials.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327523 counts factorizations of elements of A130091 using elements of A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
Cf. A001055, A022559, A022915, A027423, A091050, A124010, A317829, A327498, A327527, A336420, A336421, A336571.

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

cp's OEIS Frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

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A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).

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A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

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A336571 Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

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A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

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A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

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A337104 Number of strict chains of divisors from n! to 1 using terms of A130091 (numbers with distinct prime multiplicities).

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A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

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