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A307517
Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.
5
12, 20, 24, 28, 30, 36, 40, 42, 44, 45, 48, 52, 56, 60, 63, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 153, 154, 156, 160, 164, 165, 168
OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with at least two not necessarily distinct parts less than the largest part. The enumeration of these partitions by sum is given by A000094.
EXAMPLE
The sequence of terms together with their prime indices begins:
12: {1,1,2}
20: {1,1,3}
24: {1,1,1,2}
28: {1,1,4}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
42: {1,2,4}
44: {1,1,5}
45: {2,2,3}
48: {1,1,1,1,2}
52: {1,1,6}
56: {1,1,1,4}
60: {1,1,2,3}
63: {2,2,4}
66: {1,2,5}
68: {1,1,7}
70: {1,3,4}
72: {1,1,1,2,2}
76: {1,1,8}
MAPLE
q:= n-> (l-> add(l[i][2], i=1..nops(l)-1)>1)(sort(ifactors(n)[2])):
select(q, [$1..200])[]; # Alois P. Heinz, Apr 12 2019
MATHEMATICA
Select[Range[100], PrimeOmega[#/Power@@FactorInteger[#][[-1]]]>1&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 12 2019
STATUS
approved