A328867 - OEIS

%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