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

1,2

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}.

Links

Table of n, a(n) for n=1..60.

Example

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).

Mathematica

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

Crossrefs

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

Sequence in context: A328603 A304713 A319327 * A003277 A322833 A117287

Adjacent sequences: A319316 A319317 A319318 * A319320 A319321 A319322

Keyword

nonn

Author

Gus Wiseman, Sep 17 2018

Status

approved