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

%I A340855 #11 Apr 08 2021 03:21:12
%S A340855 3,5,7,9,11,12,13,15,17,18,19,21,23,24,25,27,29,30,31,33,35,36,37,39,
%T A340855 40,41,42,43,45,47,48,49,50,51,53,54,55,56,57,59,60,61,63,65,66,67,69,
%U A340855 70,71,72,73,75,77,78,79,80,81,83,84,85,87,89,90,91,93,95
%N A340855 Numbers that can be factored into factors > 1, the least of which is odd.
%C A340855 These are numbers that are odd or have an odd divisor 1 < d <= n/d.
%e A340855 The sequence of terms together with their prime indices begins:
%e A340855      3: {2}          27: {2,2,2}      48: {1,1,1,1,2}
%e A340855      5: {3}          29: {10}         49: {4,4}
%e A340855      7: {4}          30: {1,2,3}      50: {1,3,3}
%e A340855      9: {2,2}        31: {11}         51: {2,7}
%e A340855     11: {5}          33: {2,5}        53: {16}
%e A340855     12: {1,1,2}      35: {3,4}        54: {1,2,2,2}
%e A340855     13: {6}          36: {1,1,2,2}    55: {3,5}
%e A340855     15: {2,3}        37: {12}         56: {1,1,1,4}
%e A340855     17: {7}          39: {2,6}        57: {2,8}
%e A340855     18: {1,2,2}      40: {1,1,1,3}    59: {17}
%e A340855     19: {8}          41: {13}         60: {1,1,2,3}
%e A340855     21: {2,4}        42: {1,2,4}      61: {18}
%e A340855     23: {9}          43: {14}         63: {2,2,4}
%e A340855     24: {1,1,1,2}    45: {2,2,3}      65: {3,6}
%e A340855     25: {3,3}        47: {15}         66: {1,2,5}
%e A340855 For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).
%t A340855 Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]
%Y A340855 The version looking at greatest factor is A057716.
%Y A340855 The version for twice-balanced is A340657, with complement A340656.
%Y A340855 These factorization are counted by A340832.
%Y A340855 The complement is A340854.
%Y A340855 A033676 selects the maximum inferior divisor.
%Y A340855 A038548 counts inferior divisors, listed by A161906.
%Y A340855 A055396 selects the least prime index.
%Y A340855 - Factorizations -
%Y A340855 A001055 counts factorizations.
%Y A340855 A045778 counts strict factorizations.
%Y A340855 A316439 counts factorizations by product and length.
%Y A340855 A339890 counts factorizations of odd length.
%Y A340855 A340653 counts balanced factorizations.
%Y A340855 - Odd -
%Y A340855 A000009 counts partitions into odd parts.
%Y A340855 A024429 counts set partitions of odd length.
%Y A340855 A026424 lists numbers with odd Omega.
%Y A340855 A066208 lists Heinz numbers of partitions into odd parts.
%Y A340855 A067659 counts strict partitions of odd length (A030059).
%Y A340855 A174726 counts ordered factorizations of odd length.
%Y A340855 A332304 counts strict compositions of odd length.
%Y A340855 A340692 counts partitions of odd rank.
%Y A340855 Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.
%K A340855 nonn
%O A340855 1,1
%A A340855 _Gus Wiseman_, Feb 04 2021