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A343662
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Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.
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7
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1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2
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OFFSET
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1,4
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LINKS
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EXAMPLE
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Triangle begins:
1: 1 1
2: 1 2 1
3: 1 2 1
4: 1 3 3 1
5: 1 2 1
6: 1 4 5 2
7: 1 2 1
8: 1 4 6 4 1
9: 1 3 3 1
10: 1 4 5 2
11: 1 2 1
12: 1 6 12 10 3
13: 1 2 1
14: 1 4 5 2
15: 1 4 5 2
16: 1 5 10 10 5 1
For example, row n = 12 counts the following chains:
() (1) (2/1) (4/2/1) (12/4/2/1)
(2) (3/1) (6/2/1) (12/6/2/1)
(3) (4/1) (6/3/1) (12/6/3/1)
(4) (4/2) (12/2/1)
(6) (6/1) (12/3/1)
(12) (6/2) (12/4/1)
(6/3) (12/4/2)
(12/1) (12/6/1)
(12/2) (12/6/2)
(12/3) (12/6/3)
(12/4)
(12/6)
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MATHEMATICA
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Table[Length[Select[Reverse/@Subsets[Divisors[n], {k}], And@@Divisible@@@Partition[#, 2, 1]&]], {n, 15}, {k, 0, PrimeOmega[n]+1}]
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CROSSREFS
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A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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