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

Author

Al Black (gblack(AT)nol.net)

Keywords

• nonn
• base
• easy

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.

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.

Programs

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

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

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 *)

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

Python

from itertools import islice
def A014311_gen(): # generator of terms
    yield (n:=7)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A014311_list = list(islice(A014311_gen(),20)) # Chai Wah Wu, Mar 10 2025

Python

from math import isqrt, comb
from sympy import integer_nthroot
def A014311(n): return (1<<(r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(t:=(n>comb(m+2,3)))+1,3))-comb((k:=isqrt(b:=r+1<<1))+(b>k*(k+1)),2))+(1<<(a:=isqrt(s:=n-comb(m-(t^1)+2,3)<<1))+((s<<2)>(a<<2)*(a+1)+1))+(1<Chai Wah Wu, Mar 10 2025

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) = A367110 = 1.428591545852638123996854844400537952781688750906133068397189529775365950039... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Extensions

Extension and program by Olivier Gérard