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Number of integer partitions of n in which no two distinct parts are relatively prime.
31

%I #13 Jan 19 2021 21:54:57

%S 1,1,2,2,3,2,5,2,6,4,9,2,15,2,17,10,23,2,39,2,46,18,58,2,95,8,103,31,

%T 139,2,219,3,232,59,299,22,452,4,492,104,645,5,920,5,1006,204,1258,8,

%U 1785,21,1994,302,2442,11,3366,71,3738,497,4570,18,6253,24,6849

%N Number of integer partitions of n in which no two distinct parts are relatively prime.

%C A partition with no two distinct parts relatively prime is said to be intersecting.

%H Fausto A. C. Cariboni, <a href="/A328673/b328673.txt">Table of n, a(n) for n = 0..350</a>

%F a(n > 0) = A200976(n) + 1.

%e The a(1) = 1 through a(10) = 9 partitions (A = 10):

%e 1 2 3 4 5 6 7 8 9 A

%e 11 111 22 11111 33 1111111 44 63 55

%e 1111 42 62 333 64

%e 222 422 111111111 82

%e 111111 2222 442

%e 11111111 622

%e 4222

%e 22222

%e 1111111111

%t Table[Length[Select[IntegerPartitions[n],And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]&]],{n,0,20}]

%Y The Heinz numbers of these partitions are A328867 (strict case is A318719).

%Y The relatively prime case is A328672.

%Y The strict case is A318717.

%Y The version for non-isomorphic multiset partitions is A319752.

%Y The version for set-systems is A305843.

%Y The version involving all parts (not just distinct ones) is A200976.

%Y Cf. A000837, A202425, A305148, A305854, A306006, A316476, A326910.

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 29 2019