A328672 - OEIS

%I #16 Jan 21 2021 03:00:49

%S 0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,

%T 1,1,1,3,1,1,1,4,1,4,1,1,2,7,1,6,1,3,3,10,1,9,3,5,4,17,1,23,6,7,6,20,

%U 3,36,9,15,7,45,5,56,14,17,20,65,7,83,18,40

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

%C Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.

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

%H Fausto A. C. Cariboni, <a href="/A328672/b328672.txt">Table of n, a(n) for n = 0..400</a>

%F a(n > 0) = A202425(n) + 1.

%e Examples:

%e   a(31) = 2:         a(46) = 2:

%e     (15,10,6)          (15,15,10,6)

%e     (1^31)             (1^46)

%e   a(37) = 3:         a(47) = 7:

%e     (15,12,10)         (20,15,12)

%e     (15,10,6,6)        (21,14,12)

%e     (1^37)             (20,15,6,6)

%e   a(41) = 4:           (21,14,6,6)

%e     (20,15,6)          (15,12,10,10)

%e     (21,14,6)          (15,10,10,6,6)

%e     (15,10,10,6)       (1^47)

%e     (1^41)           a(49) = 6:

%e   a(43) = 4:           (24,15,10)

%e     (18,15,10)         (18,15,10,6)

%e     (15,12,10,6)       (15,12,12,10)

%e     (15,10,6,6,6)      (15,12,10,6,6)

%e     (1^43)             (15,10,6,6,6,6)

%e                        (1^39)

%t Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]]&]],{n,0,32}]

%Y The Heinz numbers of these partitions are A328679.

%Y The strict case is A318715.

%Y The version for non-isomorphic multiset partitions is A319759.

%Y Relatively prime partitions are A000837.

%Y Intersecting partitions are A328673.

%Y Cf. A078374, A285573, A289509, A291166, A303140, A305148, A316476, A326910, A326912.

%K nonn

%O 0,32

%A _Gus Wiseman_, Oct 29 2019