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A336865
Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.
4
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 0, 0
OFFSET
1,12
COMMENTS
Row lengths are A073093(n) = A001222(n) + 1.
EXAMPLE
The triangle begins as follows. The n-th row is shown to the right of "n:".
1: (1) 16: (1,1,1,1,1) 31: (1,1)
2: (1,1) 17: (1,1) 32: (1,1,1,1,1,1)
3: (1,1) 18: (1,2,1,1) 33: (1,2,0)
4: (1,1,1) 19: (1,1) 34: (1,2,0)
5: (1,1) 20: (1,2,1,1) 35: (1,2,0)
6: (1,2,0) 21: (1,2,0) 36: (1,2,2,2,0)
7: (1,1) 22: (1,2,0) 37: (1,1)
8: (1,1,1,1) 23: (1,1) 38: (1,2,0)
9: (1,1,1) 24: (1,2,1,2,1) 39: (1,2,0)
10: (1,2,0) 25: (1,1,1) 40: (1,2,1,2,1)
11: (1,1) 26: (1,2,0) 41: (1,1)
12: (1,2,1,1) 27: (1,1,1,1) 42: (1,3,0,0)
13: (1,1) 28: (1,2,1,1) 43: (1,1)
14: (1,2,0) 29: (1,1) 44: (1,2,1,1)
15: (1,2,0) 30: (1,3,0,0) 45: (1,2,1,1)
Row n = 72 counts the following divisors:
1 2 4 8 24 72
3 9 12
18
Row n = 1200 counts the following divisors:
1 2 4 8 16 48 400 1200
3 25 12 24 80 600
5 20 40 200
50
75
MATHEMATICA
Table[Length[Select[Divisors[n], PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]], {n, 20}, {k, 0, PrimeOmega[n]}]
CROSSREFS
A073093 gives row lengths.
A130092 gives positions of rows ending with 0.
A146291 is the version not requiring distinct prime multiplicities.
A181796 gives row sums.
A336499 is the restriction to factorial numbers.
A001222 counts prime factors, counting multiplicity.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336423 counts chains using A130091.
Sequence in context: A194341 A171905 A327406 * A262257 A144474 A298602
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Aug 06 2020
STATUS
approved