A340656 Numbers without a twice-balanced factorization.

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 42, 46, 48, 49, 51, 55, 57, 58, 60, 62, 64, 65, 66, 69, 70, 72, 74, 77, 78, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 102, 105, 106, 108, 110, 111, 112, 114, 115, 118, 119

Offset

1,1

Comments

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

(1) the number of factors;

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

(3) A001221(n).

Links

Table of n, a(n) for n=1..63.

Example

The sequence of terms together with their prime indices begins:
     4: {1,1}          33: {2,5}          64: {1,1,1,1,1,1}
     6: {1,2}          34: {1,7}          65: {3,6}
     8: {1,1,1}        35: {3,4}          66: {1,2,5}
     9: {2,2}          38: {1,8}          69: {2,9}
    10: {1,3}          39: {2,6}          70: {1,3,4}
    14: {1,4}          42: {1,2,4}        72: {1,1,1,2,2}
    15: {2,3}          46: {1,9}          74: {1,12}
    16: {1,1,1,1}      48: {1,1,1,1,2}    77: {4,5}
    21: {2,4}          49: {4,4}          78: {1,2,6}
    22: {1,5}          51: {2,7}          80: {1,1,1,1,3}
    25: {3,3}          55: {3,5}          81: {2,2,2,2}
    26: {1,6}          57: {2,8}          82: {1,13}
    27: {2,2,2}        58: {1,10}         84: {1,1,2,4}
    30: {1,2,3}        60: {1,1,2,3}      85: {3,7}
    32: {1,1,1,1,1}    62: {1,11}         86: {1,14}
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.

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], Select[facs[#], #=={}||Length[#]==PrimeNu[Times@@#]==Max[PrimeOmega/@#]&]=={}&]

Crossrefs

Positions of zeros in A340655.

The complement is A340657.

A001055 counts factorizations.

A001221 counts distinct prime factors.

A001222 counts prime factors with multiplicity.

A045778 counts strict factorizations.

A303975 counts distinct prime factors in prime indices.

A316439 counts factorizations by product and length.

Other balance-related sequences:

- A010054 counts balanced strict partitions.

- A047993 counts balanced partitions.

- A098124 counts balanced compositions.

- A106529 lists Heinz numbers of balanced partitions.

- A340596 counts co-balanced factorizations.

- A340597 lists numbers with an alt-balanced factorization.

- A340598 counts balanced set partitions.

- A340599 counts alt-balanced factorizations.

- A340600 counts unlabeled balanced multiset partitions.

- A340652 counts unlabeled twice-balanced multiset partitions.

- A340653 counts balanced factorizations.

- A340654 counts cross-balanced factorizations.

Cf. A112798, A117409, A325134, A339846, A339890, A340607, A340689, A340690.

Sequence in context: A188654 A330991 A180366 * A346041 A341614 A323644

Adjacent sequences: A340653 A340654 A340655 * A340657 A340658 A340659

Keyword

nonn

Author

Gus Wiseman, Jan 16 2021

Status

approved