A335237
Numbers whose binary indices are not a singleton nor pairwise coprime.
Original entry on oeis.org
0, 10, 11, 14, 15, 26, 27, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 74, 75, 78, 79, 90, 91, 94, 95, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 114, 115, 116
Offset: 1
The sequence of terms together with their binary expansions and binary indices begins:
0: 0 ~ {}
10: 1010 ~ {2,4}
11: 1011 ~ {1,2,4}
14: 1110 ~ {2,3,4}
15: 1111 ~ {1,2,3,4}
26: 11010 ~ {2,4,5}
27: 11011 ~ {1,2,4,5}
30: 11110 ~ {2,3,4,5}
31: 11111 ~ {1,2,3,4,5}
34: 100010 ~ {2,6}
35: 100011 ~ {1,2,6}
36: 100100 ~ {3,6}
37: 100101 ~ {1,3,6}
38: 100110 ~ {2,3,6}
39: 100111 ~ {1,2,3,6}
40: 101000 ~ {4,6}
41: 101001 ~ {1,4,6}
42: 101010 ~ {2,4,6}
43: 101011 ~ {1,2,4,6}
44: 101100 ~ {3,4,6}
The version for prime indices is
A316438.
The version for standard compositions is
A335236.
Numbers whose binary indices are pairwise coprime or a singleton:
A087087.
Non-coprime partitions are counted by
A335240.
All of the following pertain to compositions in standard order (
A066099):
Cf.
A007360,
A048793,
A051424,
A101268,
A291166,
A302569,
A326675,
A333227,
A333228,
A335235,
A335239.
-
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],!(Length[bpe[#]]==1||CoprimeQ@@bpe[#])&]
A333492
Position of first appearance of n in A271410 (LCM of binary indices).
Original entry on oeis.org
1, 2, 4, 8, 16, 6, 64, 128, 256, 18, 1024, 12, 4096, 66, 20, 32768, 65536, 258, 262144, 24, 68, 1026, 4194304, 132, 16777216, 4098, 67108864, 72, 268435456, 22, 1073741824, 2147483648, 1028, 65538, 80, 264, 68719476736, 262146, 4100, 144, 1099511627776, 70, 4398046511104
Offset: 1
The sequence together with the corresponding binary expansions and binary indices begins:
1: 1 ~ {1}
2: 10 ~ {2}
4: 100 ~ {3}
8: 1000 ~ {4}
16: 10000 ~ {5}
6: 110 ~ {2,3}
64: 1000000 ~ {7}
128: 10000000 ~ {8}
256: 100000000 ~ {9}
18: 10010 ~ {2,5}
1024: 10000000000 ~ {11}
12: 1100 ~ {3,4}
4096: 1000000000000 ~ {13}
66: 1000010 ~ {2,7}
20: 10100 ~ {3,5}
32768: 1000000000000000 ~ {16}
65536: 10000000000000000 ~ {17}
258: 100000010 ~ {2,9}
The version for prime indices is
A330225.
The version for standard compositions is
A333225.
Let q(k) be the binary indices of k:
- The elements of q(k) are row k of
A048793.
- The product of q(k) is
A096111(k).
LCM of standard compositions is
A333226.
Cf.
A000120,
A066099,
A070939,
A074761,
A076078,
A124767,
A285572,
A324837,
A328219,
A328451,
A331579,
A333227.
-
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[LCM@@bpe[n],{n,10000}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]
A330225
Position of first appearance of n in A290103 = LCM of prime indices.
Original entry on oeis.org
1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271
Offset: 1
The version for product instead of lcm is
A318871The version for standard compositions is
A333225.
The version for binary indices is
A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is
A003963(k).
- The terms of q(k) are row k of
A112798.
- The LCM of q(k) + 1 is
A328219(k).
Cf.
A000837,
A074761,
A074971,
A076078,
A285572,
A289509,
A290104,
A319333,
A324837,
A328451,
A331579,
A333226.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]
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