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Numbers with a twice-balanced factorization.
15

%I #6 Jan 19 2021 21:52:43

%S 1,2,3,5,7,11,12,13,17,18,19,20,23,24,28,29,31,36,37,40,41,43,44,45,

%T 47,50,52,53,54,56,59,61,63,67,68,71,73,75,76,79,83,88,89,92,97,98,99,

%U 100,101,103,104,107,109,113,116,117,120,124,127,131,135,136,137

%N Numbers with a twice-balanced factorization.

%C We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:

%C (1) the number of factors;

%C (2) the maximum image of A001222 over the factors;

%C (3) A001221(n).

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

%e 1: {} 29: {10} 59: {17}

%e 2: {1} 31: {11} 61: {18}

%e 3: {2} 36: {1,1,2,2} 63: {2,2,4}

%e 5: {3} 37: {12} 67: {19}

%e 7: {4} 40: {1,1,1,3} 68: {1,1,7}

%e 11: {5} 41: {13} 71: {20}

%e 12: {1,1,2} 43: {14} 73: {21}

%e 13: {6} 44: {1,1,5} 75: {2,3,3}

%e 17: {7} 45: {2,2,3} 76: {1,1,8}

%e 18: {1,2,2} 47: {15} 79: {22}

%e 19: {8} 50: {1,3,3} 83: {23}

%e 20: {1,1,3} 52: {1,1,6} 88: {1,1,1,5}

%e 23: {9} 53: {16} 89: {24}

%e 24: {1,1,1,2} 54: {1,2,2,2} 92: {1,1,9}

%e 28: {1,1,4} 56: {1,1,1,4} 97: {25}

%e The twice-balanced factorizations of 1920 (with prime indices {1,1,1,1,1,1,1,2,3}) are (8*8*30) and (8*12*20), so 1920 is in the sequence.

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Select[Range[100],Select[facs[#],#=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]!={}&]

%Y The alt-balanced version is A340597.

%Y Positions of nonzero terms in A340655.

%Y The complement is A340656.

%Y A001055 counts factorizations.

%Y A001221 counts distinct prime factors.

%Y A001222 counts prime factors with multiplicity.

%Y A045778 counts strict factorizations.

%Y A303975 counts distinct prime factors in prime indices.

%Y A316439 counts factorizations by product and length.

%Y Other balance-related sequences:

%Y - A010054 counts balanced strict partitions.

%Y - A047993 counts balanced partitions.

%Y - A098124 counts balanced compositions.

%Y - A106529 lists Heinz numbers of balanced partitions.

%Y - A340596 counts co-balanced factorizations.

%Y - A340598 counts balanced set partitions.

%Y - A340599 counts alt-balanced factorizations.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y - A340652 counts unlabeled twice-balanced multiset partitions.

%Y - A340653 counts balanced factorizations.

%Y - A340654 counts cross-balanced factorizations.

%Y Cf. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 17 2021