A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 0, 1, 4, 1, 4, 1, 0, 1, 6, 1, 1, 7, 2, 0, 1, 11, 1, 10, 2, 1, 1, 0, 1, 17, 1, 17, 4, 0, 1, 23, 2, 1, 1, 26, 4, 1, 0, 1, 37, 1, 36, 6, 2, 1, 0, 1, 53, 1, 53, 7, 2, 1, 0, 1, 70, 4, 1, 1, 77, 11, 0, 1, 103, 1, 103, 10, 4, 2, 1

Offset

1,9

Links

Table of n, a(n) for n=1..81.

Formula

T(n,k) = A078374(n/A027750(n,k)).

Example

Triangle begins:
   1
   0  1
   1  1
   1  0  1
   2  1
   2  1  0  1
   4  1
   4  1  0  1
   6  1  1
   7  2  0  1
  11  1
  10  2  1  1  0  1
  17  1
  17  4  0  1
  23  2  1  1
  26  4  1  0  1
  37  1
  36  6  2  1  0  1
  53  1
  53  7  2  1  0  1

Mathematica

Table[Length[Select[IntegerPartitions[n], And[UnsameQ@@#, GCD@@#==k]&]], {n, 20}, {k, Divisors[n]}]

Crossrefs

A regular version is A303138. Row lengths are A000005. Row sums are A000009. First column is A078374.

Cf. A000837, A027750, A289508, A289509, A303140, A303280, A319299, A319301.

Sequence in context: A367739 A165579 A130911 * A118829 A118832 A122807

Adjacent sequences: A319297 A319298 A319299 * A319301 A319302 A319303

Keyword

nonn,tabf

Author

Gus Wiseman, Sep 16 2018

Status

approved