%I #12 Jul 09 2020 16:49:57

%S 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,

%T 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,

%U 489,1,582,34,6,3,1,1,775,14,2,1,945,55,1,1,1254

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

%H Robert Israel, <a href="/A319299/b319299.txt">Table of n, a(n) for n = 1..10006</a> (rows 1 to 1358, flattened)

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

%e Triangle begins:

%e 1

%e 1 1

%e 2 1

%e 3 1 1

%e 6 1

%e 7 2 1 1

%e 14 1

%e 17 3 1 1

%e 27 2 1

%e 34 6 1 1

%e 55 1

%e 63 7 3 2 1 1

%e 100 1

%e 119 14 1 1

%e 167 6 2 1

%e 209 17 3 1 1

%e 296 1

%e 347 27 7 2 1 1

%e 489 1

%e 582 34 6 3 1 1

%p # with table A000837 obtained from that sequence

%p f:= proc(n) local D,d;

%p D:= sort(convert(numtheory:-divisors(n),list),`>`);

%p seq(A000837[d],d=D)

%p end proc:

%p map(f, [$1..60]); # _Robert Israel_, Jul 09 2020

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

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

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

%K nonn,tabf,look

%O 1,4

%A _Gus Wiseman_, Sep 16 2018