A336499 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.
1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0
Offset: 0
Examples
Triangle begins:
1
1
1 1
1 2 0
1 2 1 2 1
1 3 1 3 2 0
1 3 2 5 3 3 2 1
1 4 2 7 4 4 3 2 0
1 4 2 7 4 5 7 7 6 3 2 0
1 4 2 8 8 9 10 11 11 7 8 5 2 0
1 4 3 11 8 11 16 16 15 15 15 13 9 6 3 1
1 5 3 14 10 13 21 21 20 19 21 18 13 9 5 2 0
1 5 3 14 10 14 25 23 27 24 30 28 28 25 20 16 11 5 2 0
Row n = 7 counts the following divisors:
1 2 4 8 16 48 144 720 {}
3 9 12 24 72 360 1008
5 18 40 80 504
7 20 56 112
28
45
63
Crossrefs
A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Programs
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Mathematica
Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]
Comments