A022559 -id:A022559 - cp's OEIS frontend

A336415 Number of divisors of n! with equal prime multiplicities.

1, 1, 2, 4, 6, 10, 13, 21, 24, 28, 33, 49, 53, 85, 94, 100, 104, 168, 173, 301, 307, 317, 334, 590, 595, 603, 636, 642, 652, 1164, 1171, 2195, 2200, 2218, 2283, 2295, 2301, 4349, 4478, 4512, 4519, 8615, 8626, 16818, 16836, 16844, 17101, 33485, 33491, 33507, 33516, 33582

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number k has "equal prime multiplicities" (or is "uniform") iff its prime signature is constant, meaning that k is a power of a squarefree number.

			The a(n) uniform divisors of n for n = 1, 2, 6, 8, 30, 36 are the columns:
  1  2  6  8  30  36
     1  3  6  15  30
        2  4  10  16
        1  3   8  15
           2   6  10
           1   5   9
               4   8
               3   6
               2   5
               1   4
                   3
                   2
                   1
In 20!, the multiplicity of the third prime (5) is 4 but the multiplicity of the fourth prime (7) is 2. Hence there are 2^3 - 1 = 3 divisors with all exponents 3 (we subtract |{1}| = 1 from that count as 1 has no exponent 3). - _David A. Corneth_, Jul 27 2020

David A. Corneth, Table of n, a(n) for n = 0..9999
Jon Maiga, Computer-generated formulas for A336415, Sequence Machine.

The version for distinct prime multiplicities is A336414.
The version for nonprime perfect powers is A336416.
Uniform partitions are counted by A047966.
Uniform numbers are A072774, with nonprime terms A182853.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
Maximum divisor with distinct prime multiplicities is A327498.
Uniform divisors are counted by A327527.
Maximum uniform divisor is A336618.
1st differences are given by A048675.
Cf. A000005, A000961, A001222, A001597, A007916, A112798, A124010, A327499.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617.

Table[Length[Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); (#ex==0) || (vecmin(ex) == vecmax(ex))); \\ Michel Marcus, Jul 24 2020

a(n) = {if(n<2, return(1)); my(f = primes(primepi(n)), res = 1, t = #f); f = vector(#f, i, val(n, f[i])); for(i = 1, f[1], while(f[t] < i, t--; ); res+=(1<David A. Corneth, Jul 27 2020

a(n) = A327527(n!).

Terms a(31) and onwards from David A. Corneth, Jul 27 2020

A325276 Irregular triangle read by rows where row n is the omega-sequence of n!.

Original entry on oeis.org

1, 2, 2, 1, 4, 2, 2, 1, 5, 3, 2, 2, 1, 7, 3, 3, 1, 8, 4, 3, 2, 2, 1, 11, 4, 3, 2, 2, 1, 13, 4, 3, 2, 2, 1, 15, 4, 4, 1, 16, 5, 4, 2, 2, 1, 19, 5, 4, 2, 2, 1, 20, 6, 4, 2, 2, 1, 22, 6, 4, 2, 1, 24, 6, 5, 2, 2, 1, 28, 6, 5, 2, 2, 1, 29, 7, 5, 2, 2, 1

Offset: 0

Gus Wiseman, Apr 18 2019

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			Triangle begins:
  {}
  {}
   1
   2  2  1
   4  2  2  1
   5  3  2  2  1
   7  3  3  1
   8  4  3  2  2  1
  11  4  3  2  2  1
  13  4  3  2  2  1
  15  4  4  1
  16  5  4  2  2  1
  19  5  4  2  2  1
  20  6  4  2  2  1
  22  6  4  2  1
  24  6  5  2  2  1
  28  6  5  2  2  1
  29  7  5  2  2  1
  32  7  5  2  2  1
  33  8  5  2  2  1
  36  8  5  2  2  1
  38  8  5  2  2  1
  40  8  6  2  2  1
  41  9  6  2  2  1
  45  9  6  2  2  1
  47  9  6  2  2  1
  49  9  6  3  2  2  1
  52  9  6  3  2  2  1
  55  9  6  3  2  2  1
  56 10  6  3  2  2  1
  59 10  6  3  2  2  1

Row lengths are A325272. Row sums are A325274. Row n is row A325275(n) of A112798. Second-to-last column is A325273. Column k = 1 is A022559. Column k = 2 is A000720. Column k = 3 is A071626.
Cf. A000142, A006939, A081401, A303555, A323023, A325238, A325275, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Table[omseq[n!],{n,0,30}]

A023978 Sum of exponents in prime-power factorization of multinomial coefficient M(3n; n,n,n).

Original entry on oeis.org

0, 2, 4, 7, 7, 9, 11, 14, 12, 13, 14, 19, 18, 20, 22, 23, 19, 23, 22, 27, 25, 25, 30, 33, 30, 30, 32, 33, 31, 34, 34, 38, 33, 34, 36, 38, 34, 37, 40, 42, 39, 42, 43, 50, 49, 48, 50, 54, 49, 50, 49, 51, 51, 53, 54, 55, 51, 53, 54, 61, 57, 60, 63, 63, 56, 56, 56, 61, 60, 61, 63, 66, 61, 64, 67, 69, 67, 68

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A006480, A022559.

Cf. A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((3*n)! / n!^3); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A006480(n)).
a(n) = A022559(3*n) - 3*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023979 Sum of exponents in prime-power factorization of multinomial coefficient M(4n; n,n,n,n).

Original entry on oeis.org

0, 4, 7, 11, 12, 16, 17, 23, 21, 23, 24, 28, 27, 33, 36, 37, 33, 38, 37, 42, 41, 44, 46, 52, 48, 51, 52, 51, 50, 55, 55, 60, 55, 57, 61, 62, 58, 64, 66, 70, 65, 72, 71, 78, 78, 76, 78, 84, 79, 81, 80, 82, 82, 88, 84, 86, 82, 85, 88, 94, 90, 98, 102, 102, 93, 95, 94, 100, 100, 104, 103, 107, 102

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A008977, A022559.

Cf. A023978, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n, n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((4*n)! / n!^4); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A008977(n)).
a(n) = A022559(4*n) - 4*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023980 Sum of exponents in prime-power factorization of multinomial coefficient M(4n;2n,n,n).

Original entry on oeis.org

0, 3, 5, 8, 9, 11, 12, 17, 15, 17, 18, 20, 20, 24, 25, 26, 24, 27, 26, 30, 29, 32, 34, 37, 35, 37, 37, 37, 36, 39, 40, 44, 40, 41, 45, 45, 43, 48, 48, 51, 48, 51, 51, 56, 56, 55, 56, 60, 57, 59, 57, 59, 60, 63, 61, 63, 60, 63, 65, 68, 65, 71, 74, 73, 68, 69, 69, 73, 71, 75, 75, 77, 74, 78, 79, 81, 81

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A000897, A022559.

Cf. A023978, A023979, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[2*n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((4*n)! / ((2*n)! * n!^2)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A000897(n)).
a(n) = A022559(4*n) - A022559(2*n) - 2*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023981 Sum of exponents in prime-power factorization of multinomial coefficient M(5n; n,n,n,n,n).

Original entry on oeis.org

0, 5, 10, 14, 16, 22, 24, 31, 29, 30, 33, 40, 38, 47, 49, 51, 45, 53, 51, 57, 59, 61, 63, 71, 66, 68, 75, 73, 71, 80, 78, 86, 76, 80, 83, 86, 81, 88, 93, 97, 92, 99, 100, 105, 106, 107, 108, 117, 107, 112, 112, 115, 118, 126, 120, 124, 119, 120, 127, 134, 131, 139, 143, 141, 127, 133

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A008978, A022559.

Cf. A023978, A023979, A023980, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n, n, n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((5*n)! / n!^5); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A008978(n)).
a(n) = A022559(5*n) - 5*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023982 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;3n,n,n).

Original entry on oeis.org

0, 3, 6, 7, 9, 13, 13, 17, 17, 17, 19, 21, 20, 27, 27, 28, 26, 30, 29, 30, 34, 36, 33, 38, 36, 38, 43, 40, 40, 46, 44, 48, 43, 46, 47, 48, 47, 51, 53, 55, 53, 57, 57, 55, 57, 59, 58, 63, 58, 62, 63, 64, 67, 73, 66, 69, 68, 67, 73, 73, 74, 79, 80, 78, 71, 77, 76, 77, 79, 84, 79, 87, 82, 87, 89, 86, 89

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A001451, A022559.

Cf. A023978, A023979, A023980, A023981, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[3*n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((5*n)! / ((3*n)!*n!*n!)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001451(n)).
a(n) = A022559(5*n) - A022559(3*n) - 2*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023983 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;2n,2n,n).

Original entry on oeis.org

0, 3, 6, 8, 10, 12, 14, 19, 17, 18, 21, 24, 24, 29, 27, 29, 27, 31, 29, 33, 35, 37, 39, 41, 40, 40, 45, 45, 43, 48, 48, 54, 46, 48, 51, 52, 51, 56, 57, 59, 58, 57, 60, 61, 62, 65, 64, 69, 63, 68, 66, 69, 74, 76, 74, 78, 75, 76, 81, 82, 81, 85, 87, 83, 77, 81, 82, 84, 81, 87, 86, 93, 87, 91, 92, 93, 96

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A001459, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023984, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[2*n, 2*n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((5*n)! / ((2*n)!^2*n!)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A001459(n)).
a(n) = A022559(5*n) - 2*A022559(2*n) - A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023984 Sum of exponents in prime-power factorization of multinomial coefficient M(6n; n,n,n,n,n,n).

Original entry on oeis.org

0, 7, 13, 20, 21, 29, 33, 40, 37, 40, 43, 54, 51, 59, 64, 67, 60, 69, 67, 76, 75, 79, 85, 95, 88, 91, 96, 96, 93, 102, 102, 111, 101, 104, 108, 114, 106, 115, 121, 125, 118, 130, 130, 143, 142, 140, 146, 155, 144, 147, 148, 153, 151, 159, 157, 162, 155, 159, 164, 177, 170, 180, 186

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A008979, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023985, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n, n, n, n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((6*n)! / (n!)^6); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A008979(n)).
a(n) = A022559(6*n) - 6*A022559(n). (End)

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023985 Sum of exponents in prime-power factorization of multinomial coefficient M(6n,2n,2n,2n).

Original entry on oeis.org

0, 4, 7, 11, 12, 14, 18, 22, 19, 22, 25, 30, 30, 32, 31, 34, 33, 36, 34, 40, 39, 43, 49, 50, 49, 49, 51, 54, 51, 54, 57, 63, 56, 56, 60, 63, 61, 67, 67, 68, 67, 67, 70, 77, 76, 77, 80, 83, 78, 81, 79, 84, 85, 84, 88, 93, 89, 93, 95, 99, 95, 99, 102, 100, 94, 95, 98, 102, 98, 101, 105, 109, 104

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023986, A023987, A023990, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[2*n, 2*n, 2*n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((6*n)! / ((2*n)!)^3); \\ Amiram Eldar, Jun 11 2025

a(n) = A022559(6*n) - 3*A022559(2*n). - Amiram Eldar, Jun 11 2025

Offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

cp's OEIS Frontend

A336415 Number of divisors of n! with equal prime multiplicities.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A325276 Irregular triangle read by rows where row n is the omega-sequence of n!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A023978 Sum of exponents in prime-power factorization of multinomial coefficient M(3n; n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A023979 Sum of exponents in prime-power factorization of multinomial coefficient M(4n; n,n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A023980 Sum of exponents in prime-power factorization of multinomial coefficient M(4n;2n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A023981 Sum of exponents in prime-power factorization of multinomial coefficient M(5n; n,n,n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A023982 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;3n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A023983 Sum of exponents in prime-power factorization of multinomial coefficient M(5n;2n,2n,n).