A319299 Irregular triangle where T(n,k) is the number of integer partitions of n with GCD equal to the k-th divisor of n.

1, 1, 1, 2, 1, 3, 1, 1, 6, 1, 7, 2, 1, 1, 14, 1, 17, 3, 1, 1, 27, 2, 1, 34, 6, 1, 1, 55, 1, 63, 7, 3, 2, 1, 1, 100, 1, 119, 14, 1, 1, 167, 6, 2, 1, 209, 17, 3, 1, 1, 296, 1, 347, 27, 7, 2, 1, 1, 489, 1, 582, 34, 6, 3, 1, 1, 775, 14, 2, 1, 945, 55, 1, 1, 1254

Offset: 1

Author

Gus Wiseman, Sep 16 2018

Keywords

• nonn
• tabf
• look

Examples

			Triangle begins:
    1
    1   1
    2   1
    3   1   1
    6   1
    7   2   1   1
   14   1
   17   3   1   1
   27   2   1
   34   6   1   1
   55   1
   63   7   3   2   1   1
  100   1
  119  14   1   1
  167   6   2   1
  209  17   3   1   1
  296   1
  347  27   7   2   1   1
  489   1
  582  34   6   3   1   1
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10006 (rows 1 to 1358, flattened)

Crossrefs

A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.

Cf. A018783, A027750, A078374, A078392, A289508, A289509, A303139, A305563, A319300.

Programs

Maple

# with table A000837 obtained from that sequence
f:= proc(n) local D,d;
  D:= sort(convert(numtheory:-divisors(n),list),`>`);
  seq(A000837[d],d=D)
end proc:
map(f, [$1..60]); # Robert Israel, Jul 09 2020

Mathematica

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

Formula

T(n,k) = A000837(n/A027750(n,k)).