A340657 Numbers with a twice-balanced factorization.

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 36, 37, 40, 41, 43, 44, 45, 47, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 88, 89, 92, 97, 98, 99, 100, 101, 103, 104, 107, 109, 113, 116, 117, 120, 124, 127, 131, 135, 136, 137

Offset

1,2

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:
      1: {}            29: {10}          59: {17}
      2: {1}           31: {11}          61: {18}
      3: {2}           36: {1,1,2,2}     63: {2,2,4}
      5: {3}           37: {12}          67: {19}
      7: {4}           40: {1,1,1,3}     68: {1,1,7}
     11: {5}           41: {13}          71: {20}
     12: {1,1,2}       43: {14}          73: {21}
     13: {6}           44: {1,1,5}       75: {2,3,3}
     17: {7}           45: {2,2,3}       76: {1,1,8}
     18: {1,2,2}       47: {15}          79: {22}
     19: {8}           50: {1,3,3}       83: {23}
     20: {1,1,3}       52: {1,1,6}       88: {1,1,1,5}
     23: {9}           53: {16}          89: {24}
     24: {1,1,1,2}     54: {1,2,2,2}     92: {1,1,9}
     28: {1,1,4}       56: {1,1,1,4}     97: {25}
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.

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

The alt-balanced version is A340597.

Positions of nonzero terms in A340655.

The complement is A340656.

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.

- 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. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.

Sequence in context: A307714 A369254 A073085 * A316793 A119251 A182358

Adjacent sequences: A340654 A340655 A340656 * A340658 A340659 A340660

Keyword

nonn

Author

Gus Wiseman, Jan 17 2021

Status

approved