OFFSET
1,1
COMMENTS
These are numbers that are odd or have an odd divisor 1 < d <= n/d.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2} 27: {2,2,2} 48: {1,1,1,1,2}
5: {3} 29: {10} 49: {4,4}
7: {4} 30: {1,2,3} 50: {1,3,3}
9: {2,2} 31: {11} 51: {2,7}
11: {5} 33: {2,5} 53: {16}
12: {1,1,2} 35: {3,4} 54: {1,2,2,2}
13: {6} 36: {1,1,2,2} 55: {3,5}
15: {2,3} 37: {12} 56: {1,1,1,4}
17: {7} 39: {2,6} 57: {2,8}
18: {1,2,2} 40: {1,1,1,3} 59: {17}
19: {8} 41: {13} 60: {1,1,2,3}
21: {2,4} 42: {1,2,4} 61: {18}
23: {9} 43: {14} 63: {2,2,4}
24: {1,1,1,2} 45: {2,2,3} 65: {3,6}
25: {3,3} 47: {15} 66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).
MATHEMATICA
Select[Range[100], Function[n, n>1&&(OddQ[n]||Select[Rest[Divisors[n]], OddQ[#]&&#<=n/#&]!={})]]
CROSSREFS
The version looking at greatest factor is A057716.
These factorization are counted by A340832.
The complement is A340854.
A033676 selects the maximum inferior divisor.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2021
STATUS
approved