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A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.

%I A328867 #5 Nov 01 2019 18:41:58
%S A328867 1,2,3,4,5,7,8,9,11,13,16,17,19,21,23,25,27,29,31,32,37,39,41,43,47,
%T A328867 49,53,57,59,61,63,64,65,67,71,73,79,81,83,87,89,91,97,101,103,107,
%U A328867 109,111,113,115,117,121,125,127,128,129,131,133,137,139,147,149
%N A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.
%C A328867 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A328867 A partition with no two distinct parts relatively prime is said to be intersecting.
%e A328867 The sequence of terms together with their prime indices begins:
%e A328867     1: {}
%e A328867     2: {1}
%e A328867     3: {2}
%e A328867     4: {1,1}
%e A328867     5: {3}
%e A328867     7: {4}
%e A328867     8: {1,1,1}
%e A328867     9: {2,2}
%e A328867    11: {5}
%e A328867    13: {6}
%e A328867    16: {1,1,1,1}
%e A328867    17: {7}
%e A328867    19: {8}
%e A328867    21: {2,4}
%e A328867    23: {9}
%e A328867    25: {3,3}
%e A328867    27: {2,2,2}
%e A328867    29: {10}
%e A328867    31: {11}
%e A328867    32: {1,1,1,1,1}
%t A328867 Select[Range[100],And@@(GCD[##]>1&)@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]
%Y A328867 These are the Heinz numbers of the partitions counted by A328673.
%Y A328867 The strict case is A318719.
%Y A328867 The relatively prime version is A328868.
%Y A328867 A ranking using binary indices is A326910.
%Y A328867 The version for non-isomorphic multiset partitions is A319752.
%Y A328867 The version for divisibility (instead of relative primality) is A316476.
%Y A328867 Cf. A000837, A056239, A112798, A200976, A289509, A303283, A305843, A318715, A318716, A328336.
%K A328867 nonn
%O A328867 1,2
%A A328867 _Gus Wiseman_, Oct 30 2019