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.

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.

Links

Table of n, a(n) for n=1..89.

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.

Cf. A000005, A001221, A098859, A118914, A124010, A336422, A336500, A336571.

Sequence in context: A194341 A171905 A327406 * A262257 A144474 A298602

Adjacent sequences: A336862 A336863 A336864 * A336866 A336867 A336868

Keyword

nonn,tabf

Author

Gus Wiseman, Aug 06 2020

Status

approved