A325609 -id:A325609 - cp's OEIS frontend

A325617 Multinomial coefficient of the prime signature of n!.

1, 1, 1, 2, 4, 20, 105, 840, 3960, 51480, 675675, 10810800, 139675536, 2793510720, 58663725120, 1799020903680, 26985313555200, 782574093100800, 25992639520848000, 857757104187984000, 30021498646579440000, 1563341744336692320000, 64179292662243158400000

Offset: 0

Gus Wiseman, May 12 2019

nonn

Number of permutations of the multiset of prime factors of n!.

			The a(5) = 20 permutations of {2,2,2,3,5}:
  (22235)  (32225)  (52223)
  (22253)  (32252)  (52232)
  (22325)  (32522)  (52322)
  (22352)  (35222)  (53222)
  (22523)
  (22532)
  (23225)
  (23252)
  (23522)
  (25223)
  (25232)
  (25322)

Wikipedia, Multinomial theorem

Cf. A000142, A008480, A011371, A022559, A076934, A108731, A115627, A318762, A325272, A325273, A325508, A325543, A325544, A325609.

Table[Multinomial@@Last/@FactorInteger[n!],{n,0,15}]

a(n) = A318762(A181819(n!)).

A325614 Unsorted q-signature of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 12 2019

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
Row n lists the nonzero multiplicities in the q-factorization of n, in order of q-index. For example, row 11 is (1,1,1,1) and row 360 is (6,3,1).

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  2 1
  2 1
  3
  2 2
  2 1 1
  1 1 1 1
  3 1
  2 1 1
  3 1
  2 2 1
  4
  2 1 1
  3 2
  3 1
  3 1 1

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325615, A325660.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length/@Split[difac[n]],{n,30}]

A325615 Sorted q-signature of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 12 2019

Every positive integer has a unique q-factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
Row n is the multiset of nonzero multiplicities in the q-factorization of n. For example, row 11 is (1,1,1,1) and row 360 is (1,3,6).

			Triangle begins:
  {}
  1
  1 1
  2
  1 1 1
  1 2
  1 2
  3
  2 2
  1 1 2
  1 1 1 1
  1 3
  1 1 2
  1 3
  1 2 2
  4
  1 1 2
  2 3
  1 3
  1 1 3

Row lengths are A324923.
Row sums are A196050.
Row-maxima are A109129.
Cf. A118914, A324922, A324924, A324931, A324934, A325608, A325609, A325613, A325614, A325660.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Sort[Length/@Split[difac[n]]],{n,30}]

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A325617 Multinomial coefficient of the prime signature of n!.

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A325614 Unsorted q-signature of n.

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A325615 Sorted q-signature of n.

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