A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

1, 2, 3, 4, 5, 8, 11, 13, 15, 16, 17, 29, 31, 32, 33, 41, 43, 47, 51, 55, 59, 64, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 128, 137, 139, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 233, 241, 249, 255, 256, 257, 269, 271

Offset

1,2

Comments

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

All entries are themselves squarefree numbers (except the powers of 2).

The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

Links

Table of n, a(n) for n=1..59.

Example

The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], And[SameQ@@PrimeOmega/@primeMS[#], SquareFreeQ[Times@@primeMS[#]]]&]

Crossrefs

Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

Sequence in context: A093327 A361971 A186041 * A302591 A269571 A281493

Adjacent sequences: A322528 A322529 A322530 * A322532 A322533 A322534

Keyword

nonn

Author

Gus Wiseman, Dec 14 2018

Status

approved