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A336498
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Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.
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6
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1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 8, 11, 12, 11, 8, 4, 1, 1, 4, 8, 11, 12, 12, 12, 12, 11, 8, 4, 1, 1, 4, 8, 12, 16, 19, 20, 20, 19, 16, 12, 8, 4, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1
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OFFSET
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0,6
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COMMENTS
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Row n is row n! of A146291. Row lengths are A022559(n) + 1.
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LINKS
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Table of n, a(n) for n=0..76.
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EXAMPLE
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Triangle begins:
1
1
1 1
1 2 1
1 2 2 2 1
1 3 4 4 3 1
1 3 5 6 6 5 3 1
1 4 8 11 12 11 8 4 1
1 4 8 11 12 12 12 12 11 8 4 1
1 4 8 12 16 19 20 20 19 16 12 8 4 1
Row n = 6 counts the following divisors:
1 2 4 8 16 48 144 720
3 6 12 24 72 240
5 9 18 36 80 360
10 20 40 120
15 30 60 180
45 90
Row n = 7 counts the following divisors:
1 2 4 8 16 48 144 720 5040
3 6 12 24 72 240 1008
5 9 18 36 80 336 1680
7 10 20 40 112 360 2520
14 28 56 120 504
15 30 60 168 560
21 42 84 180 840
35 45 90 252 1260
63 126 280
70 140 420
105 210 630
315
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MATHEMATICA
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Table[Length[Select[Divisors[n!], PrimeOmega[#]==k&]], {n, 0, 10}, {k, 0, PrimeOmega[n!]}]
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CROSSREFS
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A000720 is column k = 1.
A008302 is the version for superprimorials.
A022559 gives row lengths minus one.
A027423 gives row sums.
A146291 is the generalization to non-factorials.
A336499 is the restriction to divisors in A130091.
A000142 lists factorial numbers.
A336415 counts uniform divisors of n!.
Cf. A000005, A001222, A118914, A124010, A181796, A327526, A336420.
Factorial numbers: A002982, A007489, A048656, A054991, A071626, A325272, A325617, A336414, A336415, A336416, A336418.
Sequence in context: A110535 A033941 A053256 * A102418 A317989 A106032
Adjacent sequences: A336495 A336496 A336497 * A336499 A336500 A336501
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KEYWORD
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nonn,tabf
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AUTHOR
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Gus Wiseman, Aug 03 2020
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STATUS
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approved
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