A322136 Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.

4, 8, 12, 16, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 432, 448, 480, 512, 576, 640, 648, 672, 704, 720, 768, 800, 832, 864, 896, 960, 972

Offset

1,1

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence lists all Heinz numbers of integer partitions where the number of parts is at least 1 plus half the sum of parts.

Also Heinz numbers of integer partitions that are the vertex-degrees of some hypertree. We allow no singletons in a hypertree, so 2 is not included.

Links

Table of n, a(n) for n=1..49.

Example

The sequence of partitions with Heinz numbers in the sequence begins: (11), (111), (211), (1111), (2111), (11111), (2211), (3111), (21111), (111111), (22111), (31111), (211111), (22211), (41111), (32111), (1111111).

Mathematica

Select[Range[1000], PrimeOmega[#]>=(Total[Cases[FactorInteger[#], {p_, k_}:>k*PrimePi[p]]]+2)/2&]

Crossrefs

Cf. A000569, A025065, A030019, A056156, A056239, A056503, A112798, A181821, A242414, A304382, A320922, A320923, A320924, A320925.

Sequence in context: A311393 A311394 A172412 * A212502 A071385 A329883

Adjacent sequences: A322133 A322134 A322135 * A322137 A322138 A322139

Keyword

nonn

Author

Gus Wiseman, Nov 27 2018

Status

approved