A340855 Numbers that can be factored into factors > 1, the least of which is odd.

3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95

Offset: 1

Author

Gus Wiseman, Feb 04 2021

Keywords

• nonn

Comments

These are numbers that are odd or have an odd divisor 1 < d <= n/d.

Examples

			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).
		

Crossrefs

The version looking at greatest factor is A057716.

The version for twice-balanced is A340657, with complement A340656.

These factorization are counted by A340832.

The complement is A340854.

A033676 selects the maximum inferior divisor.

A038548 counts inferior divisors, listed by A161906.

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.

A067659 counts strict partitions of odd length (A030059).

A174726 counts ordered factorizations of odd length.

A332304 counts strict compositions of odd length.

A340692 counts partitions of odd rank.

Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Programs

Mathematica

Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]