A319319 Heinz numbers of integer partitions such that every distinct submultiset has a different GCD.

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 177

Offset: 1

Author

Gus Wiseman, Sep 17 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

First differs from A304713 (Heinz numbers of pairwise indivisible partitions) at A304713(58) = 165, which is absent from this sequence because its prime indices are {2,3,5} and GCD(2,3) = GCD(2,3,5) = 1. The first term with more than two prime factors is 17719, which has prime indices {6,10,15}. The first term with more than two prime factors that is absent from A318716 is 296851, which has prime indices {12,20,30}.

Examples

			The sequence of partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (3), (4), (5), (6), (3,2), (7), (8), (9), (10), (11), (5,2), (4,3), (12), (13), (14), (15), (7,2), (16), (5,3).
		

Crossrefs

Cf. A056239, A108917, A122768, A275972, A299702, A301899, A301900, A304713, A316313, A319315, A319318, A319327.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@GCD@@@Union[Subsets[primeMS[#]]]&]