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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.

%I A319299 #12 Jul 09 2020 16:49:57
%S A319299 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 A319299 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 A319299 489,1,582,34,6,3,1,1,775,14,2,1,945,55,1,1,1254
%N 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.
%H A319299 Robert Israel, <a href="/A319299/b319299.txt">Table of n, a(n) for n = 1..10006</a> (rows 1 to 1358, flattened)
%F A319299 T(n,k) = A000837(n/A027750(n,k)).
%e A319299 Triangle begins:
%e A319299     1
%e A319299     1   1
%e A319299     2   1
%e A319299     3   1   1
%e A319299     6   1
%e A319299     7   2   1   1
%e A319299    14   1
%e A319299    17   3   1   1
%e A319299    27   2   1
%e A319299    34   6   1   1
%e A319299    55   1
%e A319299    63   7   3   2   1   1
%e A319299   100   1
%e A319299   119  14   1   1
%e A319299   167   6   2   1
%e A319299   209  17   3   1   1
%e A319299   296   1
%e A319299   347  27   7   2   1   1
%e A319299   489   1
%e A319299   582  34   6   3   1   1
%p A319299 # with table A000837 obtained from that sequence
%p A319299 f:= proc(n) local D,d;
%p A319299   D:= sort(convert(numtheory:-divisors(n),list),`>`);
%p A319299   seq(A000837[d],d=D)
%p A319299 end proc:
%p A319299 map(f, [$1..60]); # _Robert Israel_, Jul 09 2020
%t A319299 Table[Length[Select[IntegerPartitions[n],GCD@@#==k&]],{n,20},{k,Divisors[n]}]
%Y A319299 A regular version is A168532. Row lengths are A000005. Row sums are A000041. First column is A000837.
%Y A319299 Cf. A018783, A027750, A078374, A078392, A289508, A289509, A303139, A305563, A319300.
%K A319299 nonn,tabf,look
%O A319299 1,4
%A A319299 _Gus Wiseman_, Sep 16 2018