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

1,1

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.

These are the numbers whose largest proper divisor is not a prime power. - Christian Perfect, Feb 15 2026

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

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&]

Prog

(PARI) isok(n)=my(f=factor(n)[, 2]); #f && vecsum(f) >= f[#f] + 2; \\ Andrew Howroyd, Feb 15 2026

Crossrefs

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

Sequence in context: A052020 A075078 A050421 * A206552 A332832 A065201

Adjacent sequences: A307514 A307515 A307516 * A307518 A307519 A307520

Keyword

nonn

Author

Gus Wiseman, Apr 12 2019

Status

approved