login
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.
3
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}}.
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[#]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 14 2018
STATUS
approved