A340656
Numbers without a twice-balanced factorization.
Original entry on oeis.org
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
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.
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:
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A010054 counts balanced strict partitions.
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A047993 counts balanced partitions.
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A098124 counts balanced compositions.
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A106529 lists Heinz numbers of balanced partitions.
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A340596 counts co-balanced factorizations.
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A340597 lists numbers with an alt-balanced factorization.
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A340598 counts balanced set partitions.
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A340599 counts alt-balanced factorizations.
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A340600 counts unlabeled balanced multiset partitions.
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A340652 counts unlabeled twice-balanced multiset partitions.
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A340653 counts balanced factorizations.
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A340654 counts cross-balanced factorizations.
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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/@#]&]=={}&]
A340657
Numbers with a twice-balanced factorization.
Original entry on oeis.org
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
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.
The alt-balanced 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 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.
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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.
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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/@#]&]!={}&]
A340611
Number of integer partitions of n of length 2^k where k is the greatest part.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 29, 32, 34, 36, 38, 41, 42, 45, 47, 50, 52, 56, 58, 63, 66, 71, 75, 83, 88, 98, 106, 118, 128, 143, 155, 173, 188, 208, 226, 250, 270, 297, 321, 350
Offset: 0
The partitions for n = 12, 14, 16, 22, 24:
32211111 32222111 32222221 33333322 33333333
33111111 33221111 33222211 33333331 4222221111111111
33311111 33322111 4222111111111111 4322211111111111
33331111 4321111111111111 4332111111111111
4411111111111111 4422111111111111
4431111111111111
The conjugate partitions:
(8,2,2) (8,3,3) (8,4,4) (8,7,7) (8,8,8)
(8,3,1) (8,4,2) (8,5,3) (8,8,6) (16,3,3,2)
(8,5,1) (8,6,2) (16,2,2,2) (16,4,2,2)
(8,7,1) (16,3,2,1) (16,4,3,1)
(16,4,1,1) (16,5,2,1)
(16,6,1,1)
Note: A-numbers of Heinz-number sequences are in parentheses below.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (
A340609).
A168659 = partitions whose length divides their greatest part (
A340610).
A326843 = partitions of n whose length and maximum both divide n (
A326837).
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
A340689 have a factorization of length 2^max.
A340690 have a factorization of maximum 2^length.
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Table[Length[Select[IntegerPartitions[n],Length[#]==2^Max@@#&]],{n,0,30}]
A340690
Numbers with a factorization whose greatest factor is 2^k, where k is the number of factors.
Original entry on oeis.org
2, 8, 12, 16, 32, 48, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 224, 240, 256, 280, 288, 320, 336, 384, 392, 432, 448, 480, 512, 576, 640, 672, 704, 720, 768, 800, 832, 864, 896, 960, 1008, 1024, 1056, 1120, 1152, 1200, 1248, 1280, 1296, 1344
Offset: 1
The initial terms and a valid factorization of each:
2 = 2 168 = 3*7*8 512 = 2*2*2*2*32
8 = 2*4 192 = 2*2*3*16 576 = 2*2*9*16
12 = 3*4 200 = 5*5*8 640 = 2*2*10*16
16 = 4*4 224 = 4*7*8 672 = 2*3*7*16
32 = 2*2*8 240 = 5*6*8 704 = 2*2*11*16
48 = 2*3*8 256 = 2*2*4*16 720 = 3*3*5*16
64 = 2*4*8 280 = 5*7*8 768 = 2*2*2*3*32
72 = 3*3*8 288 = 2*3*3*16 800 = 2*5*5*16
80 = 2*5*8 320 = 2*2*5*16 832 = 2*2*13*16
96 = 2*6*8 336 = 6*7*8 864 = 2*3*9*16
112 = 2*7*8 384 = 2*2*6*16 896 = 2*2*14*16
120 = 3*5*8 392 = 7*7*8 960 = 2*2*15*16
128 = 2*2*2*16 432 = 3*3*3*16 1008 = 3*3*7*16
144 = 3*6*8 448 = 2*2*7*16 1024 = 2*2*2*4*32
160 = 4*5*8 480 = 2*3*5*16 1056 = 2*3*11*16
Partitions of the prescribed type are counted by
A340611.
A047993 counts balanced partitions.
A316439 counts factorizations by product and length.
A340596 counts co-balanced factorizations.
A340597 lists numbers with an alt-balanced factorization.
A340653 counts balanced factorizations.
Cf.
A106529,
A117409,
A200750,
A325134,
A340386,
A340387,
A340599,
A340607,
A340654,
A340655,
A340656,
A340657.
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facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],2^Length[#]==Max@@#&]!={}&]
Showing 1-4 of 4 results.
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