cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A084469 Position of A084467(n) in A084468(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 8, 9, 12, 10, 11, 15, 16, 13, 18, 14, 17, 20, 23, 21, 22, 19, 24, 26, 31, 27, 30, 28, 29, 25, 32, 33, 40, 34, 39, 35, 38, 36, 37, 45, 46, 41, 50, 42, 49, 43, 48, 44, 47, 54, 57, 55, 56, 51, 60, 52, 59, 53, 58, 64, 69, 65, 68, 66, 67, 61, 72, 62, 71, 63, 70
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Links

Crossrefs

Inverse permutation: A084470.

A084470 Position of A084468(n) in A084467(n).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 6, 8, 9, 11, 12, 10, 15, 17, 13, 14, 18, 16, 23, 19, 21, 22, 20, 24, 31, 25, 27, 29, 30, 28, 26, 32, 33, 35, 37, 39, 40, 38, 36, 34, 43, 45, 47, 49, 41, 42, 50, 48, 46, 44, 55, 57, 59, 51, 53, 54, 52, 60, 58, 56, 67, 69, 71, 61, 63, 65, 66, 64, 62, 72, 70, 68
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Links

Crossrefs

Inverse permutation: A084469.

A014311 Numbers with exactly 3 ones in binary expansion.

Original entry on oeis.org

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

Views

Author

Al Black (gblack(AT)nol.net)

Keywords

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

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

A084503 Binary juggling state vectors (given in decimal), associated with the three balls juggling sequence A084501.

Original entry on oeis.org

7, 7, 7, 7, 11, 7, 7, 7, 7, 7, 11, 7, 11, 7, 7, 11, 13, 7, 19, 11, 7, 19, 13, 7, 7, 7, 7, 7, 7, 7, 11, 7, 7, 11, 7, 7, 7, 11, 13, 7, 7, 19, 11, 7, 7, 19, 13, 7, 11, 7, 7, 7, 11, 7, 11, 7, 11, 13, 7, 7, 11, 13, 14, 7, 11, 21, 11, 7, 11, 21, 14, 7, 19, 11, 7, 7, 19, 11, 13, 7, 19, 13, 7, 7, 19
Offset: 0

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Links

Crossrefs

Same sequence in binary: A084504.
Cf. A084513, A084523.
A084468 gives the records.

Formula

a(0)=7, a(n)=(A084503(n-1)+(2^A084501(n))-1)/2.

A084467 Odd terms of A084457.

Original entry on oeis.org

7, 11, 13, 19, 25, 35, 21, 37, 41, 69, 49, 67, 97, 131, 73, 137, 81, 133, 161, 261, 193, 259, 145, 265, 289, 521, 321, 517, 385, 515, 273, 529, 545, 1041, 577, 1033, 641, 1029, 769, 1027, 1537, 2051, 1057, 2081, 1089, 2065, 1153, 2057, 1281, 2053, 2561, 4101
Offset: 1

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Crossrefs

Permutation of A084468. A084469(n) gives the position of a(n) in A084468(n).

Formula

a(n) = A084457(A084464(n)).

A095736 Numbers with binary weight (A000120) <= 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 48, 49, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132, 133, 134, 136
Offset: 1

Views

Author

N. J. A. Sloane, Jun 23 2009

Keywords

Comments

There are O(log^4 x) members of this sequence up to x. - Charles R Greathouse IV, Mar 29 2013

Links

Crossrefs

Cf. A000079, A000120, A048645, A018900, A084468, A014311, A001969, A000069.

Programs

Formula

Sum_{n>=2} 1/a(n) = 4.957591106549526542379494338911534917897082748621184321529665450307117309571... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

A192734 Smallest number having binary weight of 3 and n distinct prime factors.

Original entry on oeis.org

7, 21, 273, 16401, 1048593, 4295032833, 1099512676353, 9007199256838145, 302231455185132270387201, 1208944266358702884257793, 1329227995784915872903807060297121793, 1393796574908163946347162983661240005427201
Offset: 1

Views

Author

Johan Särnbratt, Jul 08 2011

Keywords

Comments

Written in binary, these numbers have exactly three 1 bits and the other bits are all 0's. This means that these numbers are of the sum of 1 plus two larger distinct powers of 2. - Alonso del Arte, Jul 08 2011
a(n) > A002110(n). [Reinhard Zumkeller, Jul 09 2011]
Sequence is not monotone: a(12) > a(13). [Charles R Greathouse IV, Jul 11 2011]

Links

Crossrefs

Cf. A084468, A001221.

Programs

  • Haskell
    a192734 n = head [x | x <- [2^u + 2^v + 1 | u <- [2..], v <- [1..u-1]],
                      a001221 x == n]
-- Reinhard Zumkeller, Jun 14 2015, Jul 09 2011
  • Mathematica
    list = {7}; For[max = 1; n = 2, n < 120,
For[m = 0, m < n,
  tal = 2*(2^n + 2^m) + 1; num = PrimeNu[tal];
  If[num > max, AppendTo[list, tal]; max = num]
  , m++], n++] (* Sarnbratt *)
A084468 = Flatten[Table[2^m + 2^n + 1, {m, 2, 80}, {n, m - 1}]]; Flatten[Table[Take[Select[A084468, PrimeNu[#] == n &], 1], {n, 10}]] (* Alonso del Arte, Jul 08 2011 *)
  • PARI
    a(n)={
  my(t);
  for(a=2,9e9,
    t=1+1<Charles R Greathouse IV, Jul 08 2011

Extensions

a(9) corrected by Charles R Greathouse IV, Jul 08 2011
a(12) from Charles R Greathouse IV, Jul 11 2011
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.