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%I A340657 #6 Jan 19 2021 21:52:43
%S A340657 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 A340657 47,50,52,53,54,56,59,61,63,67,68,71,73,75,76,79,83,88,89,92,97,98,99,
%U A340657 100,101,103,104,107,109,113,116,117,120,124,127,131,135,136,137
%N A340657 Numbers with a twice-balanced factorization.
%C A340657 We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
%C A340657 (1) the number of factors;
%C A340657 (2) the maximum image of A001222 over the factors;
%C A340657 (3) A001221(n).
%e A340657 The sequence of terms together with their prime indices begins:
%e A340657       1: {}            29: {10}          59: {17}
%e A340657       2: {1}           31: {11}          61: {18}
%e A340657       3: {2}           36: {1,1,2,2}     63: {2,2,4}
%e A340657       5: {3}           37: {12}          67: {19}
%e A340657       7: {4}           40: {1,1,1,3}     68: {1,1,7}
%e A340657      11: {5}           41: {13}          71: {20}
%e A340657      12: {1,1,2}       43: {14}          73: {21}
%e A340657      13: {6}           44: {1,1,5}       75: {2,3,3}
%e A340657      17: {7}           45: {2,2,3}       76: {1,1,8}
%e A340657      18: {1,2,2}       47: {15}          79: {22}
%e A340657      19: {8}           50: {1,3,3}       83: {23}
%e A340657      20: {1,1,3}       52: {1,1,6}       88: {1,1,1,5}
%e A340657      23: {9}           53: {16}          89: {24}
%e A340657      24: {1,1,1,2}     54: {1,2,2,2}     92: {1,1,9}
%e A340657      28: {1,1,4}       56: {1,1,1,4}     97: {25}
%e A340657 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 A340657 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340657 Select[Range[100],Select[facs[#],#=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]!={}&]
%Y A340657 The alt-balanced version is A340597.
%Y A340657 Positions of nonzero terms in A340655.
%Y A340657 The complement is A340656.
%Y A340657 A001055 counts factorizations.
%Y A340657 A001221 counts distinct prime factors.
%Y A340657 A001222 counts prime factors with multiplicity.
%Y A340657 A045778 counts strict factorizations.
%Y A340657 A303975 counts distinct prime factors in prime indices.
%Y A340657 A316439 counts factorizations by product and length.
%Y A340657 Other balance-related sequences:
%Y A340657 - A010054 counts balanced strict partitions.
%Y A340657 - A047993 counts balanced partitions.
%Y A340657 - A098124 counts balanced compositions.
%Y A340657 - A106529 lists Heinz numbers of balanced partitions.
%Y A340657 - A340596 counts co-balanced factorizations.
%Y A340657 - A340598 counts balanced set partitions.
%Y A340657 - A340599 counts alt-balanced factorizations.
%Y A340657 - A340600 counts unlabeled balanced multiset partitions.
%Y A340657 - A340652 counts unlabeled twice-balanced multiset partitions.
%Y A340657 - A340653 counts balanced factorizations.
%Y A340657 - A340654 counts cross-balanced factorizations.
%Y A340657 Cf. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.
%K A340657 nonn
%O A340657 1,2
%A A340657 _Gus Wiseman_, Jan 17 2021

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