A307517 Numbers with at least two not necessarily distinct prime factors less than the largest prime factor.

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

Author

Gus Wiseman, Apr 12 2019

Keywords

• nonn

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.

Examples

			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}
		

Crossrefs

Cf. A000094, A000245, A000961, A001222, A052126, A056239, A061395, A064989, A105441, A112798, A243055, A256617, A307516, A325196.

Programs

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&]