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

0,6

Comments

Row lengths are A022559(n) + 1.

Links

Table of n, a(n) for n=0..93.

Example

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

Mathematica

Table[Length[Select[Divisors[n!], PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]], {n, 0, 6}, {k, 0, PrimeOmega[n!]}]

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.

Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.

Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Sequence in context: A274661 A225089 A262436 * A093998 A247918 A237203

Adjacent sequences: A336496 A336497 A336498 * A336500 A336501 A336502

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 03 2020

Status

approved