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A014311
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Numbers with exactly 3 ones in binary expansion.
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64
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7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 161, 162, 164, 168, 176, 193, 194, 196, 200, 208, 224, 259, 261, 262, 265, 266, 268, 273, 274, 276, 280, 289, 290, 292, 296, 304
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OFFSET
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1,1
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COMMENTS
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Equivalently, sums of three distinct powers of 2.
Appears to give all n such that 64 is the highest power of 2 dividing A005148(n). - Benoit Cloitre, Jun 22 2002
These are numbers k such that the k-th composition in standard order has length 3. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. The sequence together with the corresponding standard compositions begins:
7: (1,1,1) 44: (2,1,3) 97: (1,5,1)
11: (2,1,1) 49: (1,4,1) 98: (1,4,2)
13: (1,2,1) 50: (1,3,2) 100: (1,3,3)
14: (1,1,2) 52: (1,2,3) 104: (1,2,4)
19: (3,1,1) 56: (1,1,4) 112: (1,1,5)
21: (2,2,1) 67: (5,1,1) 131: (6,1,1)
22: (2,1,2) 69: (4,2,1) 133: (5,2,1)
25: (1,3,1) 70: (4,1,2) 134: (5,1,2)
26: (1,2,2) 73: (3,3,1) 137: (4,3,1)
28: (1,1,3) 74: (3,2,2) 138: (4,2,2)
35: (4,1,1) 76: (3,1,3) 140: (4,1,3)
37: (3,2,1) 81: (2,4,1) 145: (3,4,1)
38: (3,1,2) 82: (2,3,2) 146: (3,3,2)
41: (2,3,1) 84: (2,2,3) 148: (3,2,3)
42: (2,2,2) 88: (2,1,4) 152: (3,1,4)
(End)
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 1.428591545852638123996854844400537952781688750906133068397189529775365950039... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022
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MATHEMATICA
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Select[Range[200], (Count[IntegerDigits[#, 2], 1] == 3)&]
nn = 8; Flatten[Table[2^i + 2^j + 2^k, {i, 2, nn}, {j, 1, i - 1}, {k, 0, j - 1}]] (* T. D. Noe, Nov 05 2013 *)
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PROG
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(Haskell)
a014311 n = a014311_list !! (n-1)
a014311_list = [2^x + 2^y + 2^z |
x <- [2..], y <- [1..x-1], z <- [0..y-1]]
(C)
unsigned hakmem175(unsigned x) {
unsigned s, o, r;
s = x & -x; r = x + s;
o = r ^ x; o = (o >> 2) / s;
return r | o;
}
if (n == 1) return 7;
(PARI) for(n=0, 10^3, if(hammingweight(n)==3, print1(n, ", "))); \\ Joerg Arndt, Mar 04 2014
(Python)
A014311_list = [2**a+2**b+2**c for a in range(2, 6) for b in range(1, a) for c in range(b)] # Chai Wah Wu, Jan 24 2021
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CROSSREFS
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Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hammingweight = 1, 2, ..., 9).
A000217(n-2) counts compositions into three parts.
A033992 lists numbers divisible by exactly three different primes.
A323024 lists numbers with exactly three different prime multiplicities.
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KEYWORD
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nonn,base,easy
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AUTHOR
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Al Black (gblack(AT)nol.net)
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EXTENSIONS
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STATUS
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approved
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