cp's OEIS Frontend

A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

%I A328672 #16 Jan 21 2021 03:00:49
%S A328672 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 A328672 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 A328672 3,36,9,15,7,45,5,56,14,17,20,65,7,83,18,40
%N A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.
%C A328672 Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.
%C A328672 A partition with no two distinct parts relatively prime is said to be intersecting.
%H A328672 Fausto A. C. Cariboni, <a href="/A328672/b328672.txt">Table of n, a(n) for n = 0..400</a>
%F A328672 a(n > 0) = A202425(n) + 1.
%e A328672 Examples:
%e A328672   a(31) = 2:         a(46) = 2:
%e A328672     (15,10,6)          (15,15,10,6)
%e A328672     (1^31)             (1^46)
%e A328672   a(37) = 3:         a(47) = 7:
%e A328672     (15,12,10)         (20,15,12)
%e A328672     (15,10,6,6)        (21,14,12)
%e A328672     (1^37)             (20,15,6,6)
%e A328672   a(41) = 4:           (21,14,6,6)
%e A328672     (20,15,6)          (15,12,10,10)
%e A328672     (21,14,6)          (15,10,10,6,6)
%e A328672     (15,10,10,6)       (1^47)
%e A328672     (1^41)           a(49) = 6:
%e A328672   a(43) = 4:           (24,15,10)
%e A328672     (18,15,10)         (18,15,10,6)
%e A328672     (15,12,10,6)       (15,12,12,10)
%e A328672     (15,10,6,6,6)      (15,12,10,6,6)
%e A328672     (1^43)             (15,10,6,6,6,6)
%e A328672                        (1^39)
%t A328672 Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]]&]],{n,0,32}]
%Y A328672 The Heinz numbers of these partitions are A328679.
%Y A328672 The strict case is A318715.
%Y A328672 The version for non-isomorphic multiset partitions is A319759.
%Y A328672 Relatively prime partitions are A000837.
%Y A328672 Intersecting partitions are A328673.
%Y A328672 Cf. A078374, A285573, A289509, A291166, A303140, A305148, A316476, A326910, A326912.
%K A328672 nonn
%O A328672 0,32
%A A328672 _Gus Wiseman_, Oct 29 2019