A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

1, 2, 4, 6, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94, 104, 106, 116, 118, 122, 124, 128, 134, 136, 142, 146, 148, 152, 158, 164, 166, 172, 178, 184, 188, 194, 202, 206, 212, 214, 218, 226, 232, 236, 244

Offset

1,2

Comments

Consists of 1 and all numbers that are even and have no odd divisor 1 < d <= n/d.

Links

Table of n, a(n) for n=1..59.

Example

The sequence of terms together with their prime indices begins:
      1: {}              44: {1,1,5}          106: {1,16}
      2: {1}             46: {1,9}            116: {1,1,10}
      4: {1,1}           52: {1,1,6}          118: {1,17}
      6: {1,2}           58: {1,10}           122: {1,18}
      8: {1,1,1}         62: {1,11}           124: {1,1,11}
     10: {1,3}           64: {1,1,1,1,1,1}    128: {1,1,1,1,1,1,1}
     14: {1,4}           68: {1,1,7}          134: {1,19}
     16: {1,1,1,1}       74: {1,12}           136: {1,1,1,7}
     20: {1,1,3}         76: {1,1,8}          142: {1,20}
     22: {1,5}           82: {1,13}           146: {1,21}
     26: {1,6}           86: {1,14}           148: {1,1,12}
     28: {1,1,4}         88: {1,1,1,5}        152: {1,1,1,8}
     32: {1,1,1,1,1}     92: {1,1,9}          158: {1,22}
     34: {1,7}           94: {1,15}           164: {1,1,13}
     38: {1,8}          104: {1,1,1,6}        166: {1,23}
For example, the factorizations of 88 are (2*2*2*11), (2*2*22), (2*4*11), (2*44), (4*22), (8*11), (88), none of which has odd minimum, so 88 is in the sequence.

Mathematica

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

Crossrefs

The version looking at greatest factor is A000079.

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

These factorization are counted by A340832.

The complement is A340855.

A033676 selects the maximum inferior divisor.

A038548 counts inferior divisors.

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.

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

Sequence in context: A138929 A334169 A180081 * A367586 A242418 A191146

Adjacent sequences: A340851 A340852 A340853 * A340855 A340856 A340857

Keyword

nonn

Author

Gus Wiseman, Feb 04 2021

Status

approved