

A340657


Numbers with a twicebalanced factorization.


15



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
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OFFSET

1,2


COMMENTS

We define a factorization of n into factors > 1 to be twicebalanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;


LINKS



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 twicebalanced 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 altbalanced version is A340597.
Positions of nonzero terms in A340655.
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 balancerelated sequences:
 A010054 counts balanced strict partitions.
 A047993 counts balanced partitions.
 A098124 counts balanced compositions.
 A106529 lists Heinz numbers of balanced partitions.
 A340596 counts cobalanced factorizations.
 A340598 counts balanced set partitions.
 A340599 counts altbalanced factorizations.
 A340600 counts unlabeled balanced multiset partitions.
 A340652 counts unlabeled twicebalanced multiset partitions.
 A340653 counts balanced factorizations.
 A340654 counts crossbalanced factorizations.
Cf. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.


KEYWORD

nonn


AUTHOR



STATUS

approved



