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%I #8 Jan 19 2021 21:52:35
%S 4,6,8,9,10,14,15,16,21,22,25,26,27,30,32,33,34,35,38,39,42,46,48,49,
%T 51,55,57,58,60,62,64,65,66,69,70,72,74,77,78,80,81,82,84,85,86,87,90,
%U 91,93,94,95,96,102,105,106,108,110,111,112,114,115,118,119
%N Numbers without 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 4: {1,1} 33: {2,5} 64: {1,1,1,1,1,1}
%e 6: {1,2} 34: {1,7} 65: {3,6}
%e 8: {1,1,1} 35: {3,4} 66: {1,2,5}
%e 9: {2,2} 38: {1,8} 69: {2,9}
%e 10: {1,3} 39: {2,6} 70: {1,3,4}
%e 14: {1,4} 42: {1,2,4} 72: {1,1,1,2,2}
%e 15: {2,3} 46: {1,9} 74: {1,12}
%e 16: {1,1,1,1} 48: {1,1,1,1,2} 77: {4,5}
%e 21: {2,4} 49: {4,4} 78: {1,2,6}
%e 22: {1,5} 51: {2,7} 80: {1,1,1,1,3}
%e 25: {3,3} 55: {3,5} 81: {2,2,2,2}
%e 26: {1,6} 57: {2,8} 82: {1,13}
%e 27: {2,2,2} 58: {1,10} 84: {1,1,2,4}
%e 30: {1,2,3} 60: {1,1,2,3} 85: {3,7}
%e 32: {1,1,1,1,1} 62: {1,11} 86: {1,14}
%e For example, the factorizations of 48 with (2) and (3) equal are: (2*2*2*6), (2*2*3*4), (2*4*6), (3*4*4), but since none of these has length 2, the sequence contains 48.
%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 Positions of zeros in A340655.
%Y The complement is A340657.
%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 - A340597 lists numbers with an alt-balanced factorization.
%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. A112798, A117409, A325134, A339846, A339890, A340607, A340689, A340690.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jan 16 2021
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