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Numbers that cannot be factored into factors > 1, the least of which is odd.
20

%I #10 Apr 09 2021 09:41:21

%S 1,2,4,6,8,10,14,16,20,22,26,28,32,34,38,44,46,52,58,62,64,68,74,76,

%T 82,86,88,92,94,104,106,116,118,122,124,128,134,136,142,146,148,152,

%U 158,164,166,172,178,184,188,194,202,206,212,214,218,226,232,236,244

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

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

%e The sequence of terms together with their prime indices begins:

%e 1: {} 44: {1,1,5} 106: {1,16}

%e 2: {1} 46: {1,9} 116: {1,1,10}

%e 4: {1,1} 52: {1,1,6} 118: {1,17}

%e 6: {1,2} 58: {1,10} 122: {1,18}

%e 8: {1,1,1} 62: {1,11} 124: {1,1,11}

%e 10: {1,3} 64: {1,1,1,1,1,1} 128: {1,1,1,1,1,1,1}

%e 14: {1,4} 68: {1,1,7} 134: {1,19}

%e 16: {1,1,1,1} 74: {1,12} 136: {1,1,1,7}

%e 20: {1,1,3} 76: {1,1,8} 142: {1,20}

%e 22: {1,5} 82: {1,13} 146: {1,21}

%e 26: {1,6} 86: {1,14} 148: {1,1,12}

%e 28: {1,1,4} 88: {1,1,1,5} 152: {1,1,1,8}

%e 32: {1,1,1,1,1} 92: {1,1,9} 158: {1,22}

%e 34: {1,7} 94: {1,15} 164: {1,1,13}

%e 38: {1,8} 104: {1,1,1,6} 166: {1,23}

%e 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.

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

%Y The version looking at greatest factor is A000079.

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

%Y These factorization are counted by A340832.

%Y The complement is A340855.

%Y A033676 selects the maximum inferior divisor.

%Y A038548 counts inferior divisors.

%Y A055396 selects the least prime index.

%Y - Factorizations -

%Y A001055 counts factorizations.

%Y A045778 counts strict factorizations.

%Y A316439 counts factorizations by product and length.

%Y A339890 counts factorizations of odd length.

%Y A340653 counts balanced factorizations.

%Y - Odd -

%Y A000009 counts partitions into odd parts.

%Y A024429 counts set partitions of odd length.

%Y A026424 lists numbers with odd Omega.

%Y A066208 lists Heinz numbers of partitions into odd parts.

%Y A067659 counts strict partitions of odd length (A030059).

%Y A174726 counts ordered factorizations of odd length.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 04 2021