login
A363488
Even numbers whose prime factorization has at least as many 2's as non-2's.
11
2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 84, 86, 88, 92, 94, 96, 100, 104, 106, 112, 116, 118, 120, 122, 124, 128, 132, 134, 136, 140, 142, 144, 146, 148, 152, 156, 158, 160
OFFSET
1,1
COMMENTS
The multiset of prime factors of n is row n of A027746.
Also numbers whose prime factors have low median 2, where the low median (see A124943) is either the middle part (for odd length), or the least of the two middle parts (for even length).
EXAMPLE
The terms together with their prime indices begin:
2: {1} 34: {1,7} 72: {1,1,1,2,2}
4: {1,1} 36: {1,1,2,2} 74: {1,12}
6: {1,2} 38: {1,8} 76: {1,1,8}
8: {1,1,1} 40: {1,1,1,3} 80: {1,1,1,1,3}
10: {1,3} 44: {1,1,5} 82: {1,13}
12: {1,1,2} 46: {1,9} 84: {1,1,2,4}
14: {1,4} 48: {1,1,1,1,2} 86: {1,14}
16: {1,1,1,1} 52: {1,1,6} 88: {1,1,1,5}
20: {1,1,3} 56: {1,1,1,4} 92: {1,1,9}
22: {1,5} 58: {1,10} 94: {1,15}
24: {1,1,1,2} 60: {1,1,2,3} 96: {1,1,1,1,1,2}
26: {1,6} 62: {1,11} 100: {1,1,3,3}
28: {1,1,4} 64: {1,1,1,1,1,1} 104: {1,1,1,6}
32: {1,1,1,1,1} 68: {1,1,7} 106: {1,16}
MATHEMATICA
Select[Range[100], EvenQ[#]&&PrimeOmega[#]<=2*FactorInteger[#][[1, 2]]&]
CROSSREFS
Partitions of this type are counted by A027336.
The case without high median > 1 is A072978.
For mode instead of median we have A360015, high A360013.
Positions of 1's in A363941.
For mean instead of median we have A363949, high A000079.
The high version is A364056, positions of 1's in A363942.
A067538 counts partitions with integer mean, ranks A316413.
A112798 lists prime indices, length A001222, sum A056239.
A124943 counts partitions by low median, high A124944.
A363943 gives low mean of prime indices, triangle A363945.
Sequence in context: A249124 A322405 A360015 * A118081 A152483 A330688
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 06 2023
STATUS
approved