login
Numbers that can be factored into factors > 1, the least of which is odd.
19

%I #11 Apr 08 2021 03:21:12

%S 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 40,41,42,43,45,47,48,49,50,51,53,54,55,56,57,59,60,61,63,65,66,67,69,

%U 70,71,72,73,75,77,78,79,80,81,83,84,85,87,89,90,91,93,95

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

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

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

%e 3: {2} 27: {2,2,2} 48: {1,1,1,1,2}

%e 5: {3} 29: {10} 49: {4,4}

%e 7: {4} 30: {1,2,3} 50: {1,3,3}

%e 9: {2,2} 31: {11} 51: {2,7}

%e 11: {5} 33: {2,5} 53: {16}

%e 12: {1,1,2} 35: {3,4} 54: {1,2,2,2}

%e 13: {6} 36: {1,1,2,2} 55: {3,5}

%e 15: {2,3} 37: {12} 56: {1,1,1,4}

%e 17: {7} 39: {2,6} 57: {2,8}

%e 18: {1,2,2} 40: {1,1,1,3} 59: {17}

%e 19: {8} 41: {13} 60: {1,1,2,3}

%e 21: {2,4} 42: {1,2,4} 61: {18}

%e 23: {9} 43: {14} 63: {2,2,4}

%e 24: {1,1,1,2} 45: {2,2,3} 65: {3,6}

%e 25: {3,3} 47: {15} 66: {1,2,5}

%e For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).

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

%Y The version looking at greatest factor is A057716.

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

%Y These factorization are counted by A340832.

%Y The complement is A340854.

%Y A033676 selects the maximum inferior divisor.

%Y A038548 counts inferior divisors, listed by A161906.

%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 A332304 counts strict compositions of odd length.

%Y A340692 counts partitions of odd rank.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 04 2021