A320340 Heinz numbers of double-free integer partitions.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset

1,2

Comments

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

An integer partition is double-free if no part is twice any other part.

Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Eric Weisstein's World of Mathematics, Double-Free Set

Example

The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (41), (32), (1111), (7), (8), (311), (51), (9), (33), (61), (222), (411).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Intersection[primeMS[#], 2*primeMS[#]]=={}&]

Crossrefs

Cf. A018819, A056239, A087897, A101417, A112798, A120641, A276431, A305148, A319613, A323092, A323093.

Sequence in context: A043091 A031943 A023800 * A350838 A364347 A228869

Adjacent sequences: A320337 A320338 A320339 * A320341 A320342 A320343

Keyword

nonn

Author

Gus Wiseman, Jan 07 2019

Status

approved