A014311 Numbers with exactly 3 ones in binary expansion.

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

Offset

1,1

Comments

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

From Gus Wiseman, Oct 05 2020: (Start)

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)

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Stephen Morley, HAKMEM Item 175 (Gosper).

Tilman Piesk, First 56 elements in a tetrahedral array.

Formula

A000120(a(n)) = 3. - Reinhard Zumkeller, May 03 2012

Start with A084468. If n is in sequence, then 2n is too. - Ralf Stephan, Aug 16 2013

a(n+1) = A057168(a(n)). - M. F. Hasler, Aug 27 2014

a(n) = 2^A056558(n-1) + 2^A194848(n-1) + 2^A194847(n-1). - Ridouane Oudra, Sep 06 2020

Sum_{n>=1} 1/a(n) = 1.428591545852638123996854844400537952781688750906133068397189529775365950039... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Mathematica

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 *)

Prog

(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]]
-- Reinhard Zumkeller, May 03 2012
(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;
}
unsigned A014311(int n) {
    if (n == 1) return 7;
    return hakmem175(A014311(n - 1));
}  // Peter Luschny, Jan 01 2014
(PARI) for(n=0, 10^3, if(hammingweight(n)==3, print1(n, ", "))); \\ Joerg Arndt, Mar 04 2014
(PARI) print1(t=7); for(i=2, 50, print1(", "t=A057168(t))) \\ M. F. Hasler, Aug 27 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

Crossrefs

Cf. A038465 (base 3), A038471 (base 4), A038475 (base 5).

Cf. A081091 (primes), A212190 (squares), A212192 (triangular numbers), A173589 (Fibbinary).

Cf. A057168.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hammingweight = 1, 2, ..., 9).

Cf. A056558, A194848, A194847.

A000217(n-2) counts compositions into three parts.

A001399(n-3) = A069905(n) = A211540(n+2) counts the unordered case.

A001399(n-6) = A069905(n-3) = A211540(n-1) counts the unordered strict case.

A001399(n-6)*6 = A069905(n-3)*6 = A211540(n-1)*6 counts the strict case.

A014612 is an unordered version, with strict case A007304.

A337453 is the strict case.

A337461 counts the coprime case.

A033992 lists numbers divisible by exactly three different primes.

A323024 lists numbers with exactly three different prime multiplicities.

Cf. A220377, A307534, A337459, A337460, A337561, A337603, A337604.

Sequence in context: A271499 A235336 A075930 * A337459 A245178 A287161

Adjacent sequences: A014308 A014309 A014310 * A014312 A014313 A014314

Keyword

nonn,base,easy

Author

Al Black (gblack(AT)nol.net)

Extensions

Extension and program by Olivier Gérard

Status

approved