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A340656 Numbers without a twice-balanced factorization.

%I A340656 #8 Jan 19 2021 21:52:35
%S A340656 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 A340656 51,55,57,58,60,62,64,65,66,69,70,72,74,77,78,80,81,82,84,85,86,87,90,
%U A340656 91,93,94,95,96,102,105,106,108,110,111,112,114,115,118,119
%N A340656 Numbers without a twice-balanced factorization.
%C A340656 We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
%C A340656 (1) the number of factors;
%C A340656 (2) the maximum image of A001222 over the factors;
%C A340656 (3) A001221(n).
%e A340656 The sequence of terms together with their prime indices begins:
%e A340656      4: {1,1}          33: {2,5}          64: {1,1,1,1,1,1}
%e A340656      6: {1,2}          34: {1,7}          65: {3,6}
%e A340656      8: {1,1,1}        35: {3,4}          66: {1,2,5}
%e A340656      9: {2,2}          38: {1,8}          69: {2,9}
%e A340656     10: {1,3}          39: {2,6}          70: {1,3,4}
%e A340656     14: {1,4}          42: {1,2,4}        72: {1,1,1,2,2}
%e A340656     15: {2,3}          46: {1,9}          74: {1,12}
%e A340656     16: {1,1,1,1}      48: {1,1,1,1,2}    77: {4,5}
%e A340656     21: {2,4}          49: {4,4}          78: {1,2,6}
%e A340656     22: {1,5}          51: {2,7}          80: {1,1,1,1,3}
%e A340656     25: {3,3}          55: {3,5}          81: {2,2,2,2}
%e A340656     26: {1,6}          57: {2,8}          82: {1,13}
%e A340656     27: {2,2,2}        58: {1,10}         84: {1,1,2,4}
%e A340656     30: {1,2,3}        60: {1,1,2,3}      85: {3,7}
%e A340656     32: {1,1,1,1,1}    62: {1,11}         86: {1,14}
%e A340656 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 A340656 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340656 Select[Range[100],Select[facs[#],#=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]=={}&]
%Y A340656 Positions of zeros in A340655.
%Y A340656 The complement is A340657.
%Y A340656 A001055 counts factorizations.
%Y A340656 A001221 counts distinct prime factors.
%Y A340656 A001222 counts prime factors with multiplicity.
%Y A340656 A045778 counts strict factorizations.
%Y A340656 A303975 counts distinct prime factors in prime indices.
%Y A340656 A316439 counts factorizations by product and length.
%Y A340656 Other balance-related sequences:
%Y A340656 - A010054 counts balanced strict partitions.
%Y A340656 - A047993 counts balanced partitions.
%Y A340656 - A098124 counts balanced compositions.
%Y A340656 - A106529 lists Heinz numbers of balanced partitions.
%Y A340656 - A340596 counts co-balanced factorizations.
%Y A340656 - A340597 lists numbers with an alt-balanced factorization.
%Y A340656 - A340598 counts balanced set partitions.
%Y A340656 - A340599 counts alt-balanced factorizations.
%Y A340656 - A340600 counts unlabeled balanced multiset partitions.
%Y A340656 - A340652 counts unlabeled twice-balanced multiset partitions.
%Y A340656 - A340653 counts balanced factorizations.
%Y A340656 - A340654 counts cross-balanced factorizations.
%Y A340656 Cf. A112798, A117409, A325134, A339846, A339890, A340607, A340689, A340690.
%K A340656 nonn
%O A340656 1,1
%A A340656 _Gus Wiseman_, Jan 16 2021