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A322136
Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.
3
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.
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&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 27 2018
STATUS
approved