%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