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