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 1-5 of 5 results.

A181742 Let A181741(n)=2^(t(n))-2^(k(n))-1, where k(n)>=1, t(n)>=k(n)+1. Then a(n)=t(n).

Original entry on oeis.org

3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 24
Offset: 1

Views

Author

Vladimir Shevelev, Nov 08 2010

Keywords

Crossrefs

Cf. A181741, A181743.

Programs

  • Mathematica
    f[n_] := IntegerExponent[n + 2^IntegerExponent[n, 2], 2]; f/@ (Select[Table[2^t-2^k-1, {t, 1, 20}, {k, 1, t-1}] // Flatten // Union, PrimeQ] + 1) (* Amiram Eldar, Dec 17 2018 after Jean-François Alcover at A181741 *)
  • PARI
    listt(nn) = {for (n=3, nn, forstep(k=n-1, 1, -1, if (isprime(2^n-2^k-1), print1(n, ", "));););} \\ Michel Marcus, Dec 17 2018

Extensions

Corrected and extended by Michel Marcus, Dec 17 2018

A181743 The exponent k which defines A181741(n) = 2^t-2^k-1.

Original entry on oeis.org

2, 1, 3, 2, 1, 3, 1, 5, 4, 2, 1, 7, 6, 5, 4, 2, 7, 5, 3, 1, 5, 2, 1, 3, 9, 7, 4, 2, 1, 11, 13, 10, 8, 6, 1, 11, 7, 4, 11, 3, 17, 14, 13, 9, 8, 6, 5, 4, 2, 11, 19, 18, 17, 14, 12, 11, 10, 9, 7, 4, 2, 1, 17, 9, 7, 3, 16, 10, 5, 4, 1, 21, 15, 13, 10, 5, 4, 1, 13, 9, 2
Offset: 1

Views

Author

Vladimir Shevelev, Nov 08 2010

Keywords

Crossrefs

Cf. A181741, A181742.

Programs

  • Mathematica
    IntegerExponent[Select[Table[2^t-2^k-1, {t, 1, 20}, {k, 1, t-1}] // Flatten // Union, PrimeQ] + 1, 2] (* Amiram Eldar, Dec 17 2018 after Jean-François Alcover at A181741 *)
  • PARI
    listk(nn) = {for (n=3, nn, forstep(k=n-1, 1, -1, if (isprime(2^n-2^k-1), print1(k, ", "));););} \\ Michel Marcus, Dec 17 2018
  • Python
    from itertools import count, islice
from sympy import isprime
def A181743_gen(): # generator of terms
    m = 2
    for t in count(1):
        r=1<>=1
        m<<=1
A181743_list=list(islice(A181743_gen(),30)) # Chai Wah Wu, Jul 08 2022

Formula

k = A007814(A181741(n)+1). [R. J. Mathar, Nov 18 2010]

Extensions

Terms equivalent to insertions in A181741 inserted by R. J. Mathar, Nov 18 2010
More terms from Michel Marcus, Dec 17 2018

A089633 Numbers having no more than one 0 in their binary representation.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 23, 27, 29, 30, 31, 47, 55, 59, 61, 62, 63, 95, 111, 119, 123, 125, 126, 127, 191, 223, 239, 247, 251, 253, 254, 255, 383, 447, 479, 495, 503, 507, 509, 510, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1022, 1023
Offset: 0

Views

Author

Reinhard Zumkeller, Jan 01 2004

Keywords

Comments

Complement of A158582. - Reinhard Zumkeller, Apr 16 2009
Also union of A168604 and A030130. - Douglas Latimer, Jul 19 2012
Numbers of the form 2^t - 2^k - 1, 0 <= k < t.
n is in the sequence if and only if 2*n+1 is in the sequence. - Robert Israel, Dec 14 2018
Also the least binary rank of a strict integer partition of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 24 2024

Examples

			From _Tilman Piesk_, May 09 2012: (Start)
This may also be viewed as a triangle:             In binary:
                  0                                         0
               1     2                                 01       10
             3    5    6                          011      101      110
           7   11   13   14                  0111     1011     1101     1110
        15   23   27   29   30          01111    10111    11011    11101    11110
      31  47   55   59   61   62
   63   95  111  119  123  125  126
Left three diagonals are A000225,  A055010, A086224. Right diagonal is A000918. Central column is A129868. Numbers in row n (counted from 0) have n binary 1s. (End)
From _Gus Wiseman_, May 24 2024: (Start)
The terms together with their binary expansions and binary indices begin:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
  11:   1011 ~ {1,2,4}
  13:   1101 ~ {1,3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  23:  10111 ~ {1,2,3,5}
  27:  11011 ~ {1,2,4,5}
  29:  11101 ~ {1,3,4,5}
  30:  11110 ~ {2,3,4,5}
  31:  11111 ~ {1,2,3,4,5}
  47: 101111 ~ {1,2,3,4,6}
  55: 110111 ~ {1,2,3,5,6}
  59: 111011 ~ {1,2,4,5,6}
  61: 111101 ~ {1,3,4,5,6}
  62: 111110 ~ {2,3,4,5,6}
(End)

Links

Crossrefs

Cf. A000120, A007088, A011371, A014132, A023416, A023532, A029931.
Cf. A181741 (primes), union of A081118 and A000918, apart from initial -1.
Cf. A065442, A160502, A265705.
For least binary index (instead of rank) we have A001511.
Applying A019565 (Heinz number of binary indices) gives A077011.
For greatest binary index we have A029837 or A070939, opposite A070940.
Row minima of A118462 (binary ranks of strict partitions).
For sum instead of minimum we have A372888, non-strict A372890.
A000009 counts strict partitions, ranks A005117.
A048675 gives binary rank of prime indices, distinct A087207.
A048793 lists binary indices, product A096111, reverse A272020.
A277905 groups all positive integers by binary rank of prime indices.
Cf. A000041, A005940, A018819, A147655, A225620, A246868.

Programs

  • Haskell
    a089633 n = a089633_list !! (n-1)
a089633_list = [2 ^ t - 2 ^ k - 1 | t <- [1..], k <- [t-1,t-2..0]]
-- Reinhard Zumkeller, Feb 23 2012
  • Maple
    seq(seq(2^a-1-2^b,b=a-1..0,-1),a=1..11); # Robert Israel, Dec 14 2018
  • Mathematica
    fQ[n_] := DigitCount[n, 2, 0] < 2; Select[ Range[0, 2^10], fQ] (* Robert G. Wilson v, Aug 02 2012 *)
  • PARI
    {insq(n) = local(dd, hf, v); v=binary(n);hf=length(v);dd=sum(i=1,hf,v[i]);if(dd<=hf-2,-1,1)}
{for(w=0,1536,if(insq(w)>=0,print1(w,", ")))}
\\ Douglas Latimer, May 07 2013
  • PARI
    isoka(n) = #select(x->(x==0), binary(n)) <= 1; \\ Michel Marcus, Dec 14 2018
  • Python
    from itertools import count, islice
def A089633_gen(): # generator of terms
    return ((1<A089633_list = list(islice(A089633_gen(),30)) # Chai Wah Wu, Feb 10 2023
  • Python
    from math import isqrt, comb
def A089633(n): return (1<<(a:=(isqrt((n<<3)+1)-1>>1)+1))-(1<Chai Wah Wu, Dec 19 2024

Formula

A023416(a(n)) <= 1; A023416(a(n)) = A023532(n-2) for n>1;
A000120(a(u)) <= A000120(a(v)) for uA000120(a(n)) = A003056(n).
a(0)=0, n>0: a(n+1) = Min{m>n: BinOnes(a(n))<=BinOnes(m)} with BinOnes=A000120.
If m = floor((sqrt(8*n+1) - 1) / 2), then a(n) = 2^(m+1) - 2^(m*(m+3)/2 - n) - 1. - Carl R. White, Feb 10 2009
A029931(a(n)) = n and A029931(m) != n for m < a(n). - Reinhard Zumkeller, Feb 28 2014
A265705(a(n),k) = A265705(a(n),a(n)-k), k = 0 .. a(n). - Reinhard Zumkeller, Dec 15 2015
a(A014132(n)-1) = 2*a(n-1)+1 for n >= 1. - Robert Israel, Dec 14 2018
Sum_{n>=1} 1/a(n) = A065442 + A160502 = 3.069285887459... . - Amiram Eldar, Jan 09 2024
A019565(a(n)) = A077011(n). - Gus Wiseman, May 24 2024

A081118 Triangle of first n numbers per row having exactly n 1's in binary representation.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 23, 27, 29, 31, 47, 55, 59, 61, 63, 95, 111, 119, 123, 125, 127, 191, 223, 239, 247, 251, 253, 255, 383, 447, 479, 495, 503, 507, 509, 511, 767, 895, 959, 991, 1007, 1015, 1019, 1021, 1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 06 2003

Keywords

Comments

T(n,n) = A036563(n+1) = 2^(n+1) - 3.
Numbers of the form 2^t - 2^k - 1, 1 <= k < t.

Examples

			Triangle begins:
.......... 1 ......... ................ 1
........ 3...5 ....... .............. 11 101
...... 7..11..13 ..... .......... 111 1011 1101
... 15..23..27..29 ... ...... 1111 10111 11011 11101
. 31..47..55..59..61 . . 11111 101111 110111 111011 111101.

Links

Crossrefs

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462, A038463.
Cf. A181741 (primes), A208083, subsequence of A089633.
Cf. A131094.

Programs

  • Haskell
    a081118 n k = a081118_tabl !! (n-1) !! (k-1)
a081118_row n = a081118_tabl !! (n-1)
a081118_tabl  = iterate
   (\row -> (map ((+ 1) . (* 2)) row) ++ [4 * (head row) + 1]) [1]
a081118_list = concat a081118_tabl
-- Reinhard Zumkeller, Feb 23 2012
  • Mathematica
    Table[2^(n+1)-2^(n-k+1)-1,{n,10},{k,n}]//Flatten (* Harvey P. Dale, Apr 09 2020 *)

Formula

T(n, k) = 2^(n+1) - 2^(n-k+1) - 1, 1<=k<=n.
a(n) = (2^A002260(n)-1)*2^A004736(n)-1; a(n)=(2^i-1)*2^j-1, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 04 2013

A208083 Number of primes of the form 2^n - 2^k - 1, 1 <= k < n.

Original entry on oeis.org

0, 0, 2, 3, 2, 4, 0, 5, 4, 3, 1, 5, 1, 5, 0, 3, 2, 9, 1, 12, 4, 5, 0, 7, 1, 2, 0, 1, 5, 4, 0, 8, 5, 1, 1, 9, 0, 6, 0, 7, 1, 6, 0, 4, 7, 2, 1, 10, 3, 3, 1, 2, 1, 6, 0, 4, 3, 0, 1, 8, 3, 4, 0, 3, 1, 8, 1, 2, 2, 3, 0, 9, 1, 5, 2, 5, 8, 3, 0, 10, 3, 0, 2, 4, 4, 6
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 23 2012

Keywords

Comments

Number of primes in (n-1)-st row of the triangle in A081118;
a(A138290(n)+1) = 0;
for n >= 0: a(A208091(n)) = n and a(m) <> n for m < A208091(n).

Examples

			n _ A208083(n) ________________ (n-1)-st row of A081118 _________
5   #{23,29} = 2                [15,23,27,29]
6   #{31,47,59,61} = 4          [31,47,55,59,61]
7   #{} = 0                     [63,95,111,119,123,125]
8   #{127,191,223,239,251} = 5  [127,191,223,239,247,251,253]
9   #{383,479,503,509} = 4      [255,383,447,479,495,503,507,509]

Links

Crossrefs

Cf. A010051, A081118, A138290, A181741, A208091.

Programs

  • Haskell
    a208083 = sum . map a010051 . a081118_row
  • Maple
    f:= n -> nops(select(k -> isprime(2^n-2^k-1),[$1..n-1])):
map(f, [$1..100]); # Robert Israel, Jun 12 2018
  • Mathematica
    a[n_] := Module[{m = 2^n - 1, cnt = 0}, For[ k = 1, k < n, k++, If[PrimeQ[m - 2^k], cnt++]]; cnt]; Table[a[n], {n, 2, 86}] (* Jean-François Alcover, Sep 12 2013 *)
  • PARI
    a(n)=sum(k=1,n-1,ispseudoprime(2^n-2^k-1)) \\ Charles R Greathouse IV, Sep 12 2013

Formula

a(n) = Sum_{k=1..n-1} A010051(A081118(n-1,k)).
Showing 1-5 of 5 results.

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