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

Author

Gus Wiseman, Nov 27 2018

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

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

Programs

Mathematica

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