A301899 Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109

Offset

1,2

Comments

An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=1..66.

Formula

Intersection of A299702 and A005117.

Example

42 is the Heinz number of (4,2,1) which is strict and knapsack, so is in the sequence. 45 is the Heinz number of (3,2,2) which is knapsack but not strict, so is not in the sequence. 30 is the Heinz number of (3,2,1) which is strict but not knapsack, so is not in the sequence.
Sequence of strict knapsack partitions begins: (), (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (8), (42), (51), (9), (61).

Mathematica

wt[n_]:=If[n===1, 0, Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]]];
Select[Range[100], SquareFreeQ[#]&&UnsameQ@@wt/@Divisors[#]&]

Crossrefs

Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301900.

Sequence in context: A319315 A325467 A325779 * A325398 A325399 A167171

Adjacent sequences: A301896 A301897 A301898 * A301900 A301901 A301902

Keyword

nonn

Author

Gus Wiseman, Mar 28 2018

Status

approved