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 1-6 of 6 results.

A268692 Numbers k such that 2^(k-1)*(2^k - 1) + 1 is prime (see A134169).

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 13, 19, 45, 46, 58, 141, 271, 336, 562, 601, 1128, 1635, 2718, 2920, 3933, 4351, 4729, 6556, 8349, 10851, 32641, 34039, 41050, 63732, 64738, 68173, 88690
Offset: 1

Views

Author

Jeppe Stig Nielsen, Feb 11 2016

Keywords

Comments

The intersection of this sequence with A000043 gives 2, 3, 13, 19, ... which are the indices corresponding to primes just next to perfect numbers (A000396), see A061644.
There are prime members of this sequence (271, 601, 4729, ...) which are not in A000043.
a(30) > 50000. All the primes corresponding to terms up to a(29) have been certified by the PFGW software performing the Brillhart-Lehmer-Selfridge N-1 test. - Giovanni Resta, Apr 11 2016
a(30)-a(32) terms have been certified by the PFGW software performing the Brillhart-Lehmer-Selfridge N-1 test. - Jorge Coveiro, Oct 29 2023
a(33) term has been certified by the PFGW software performing the Brillhart-Lehmer-Selfridge N-1 test. - Jorge Coveiro, Mar 08 2024

Crossrefs

Cf. A134169, A061644, A000043, A000396, A006516.

Programs

  • PARI
    for(n=0,10^5,ispseudoprime(2^(n-1)*(2^n-1)+1) && print1(n,", "))

Extensions

a(27)-a(29) from Giovanni Resta, Apr 11 2016
a(30)-a(32) from Jorge Coveiro, Oct 29 2023
a(33) from Jorge Coveiro, Mar 08 2024

A224195 Ordered sequence of numbers of form (2^n - 1)*2^m + 1 where n >= 1, m >= 1.

Original entry on oeis.org

3, 5, 7, 9, 13, 15, 17, 25, 29, 31, 33, 49, 57, 61, 63, 65, 97, 113, 121, 125, 127, 129, 193, 225, 241, 249, 253, 255, 257, 385, 449, 481, 497, 505, 509, 511, 513, 769, 897, 961, 993, 1009, 1017, 1021, 1023, 1025, 1537, 1793, 1921, 1985, 2017, 2033, 2041, 2045, 2047
Offset: 1

Views

Author

Brad Clardy, Apr 01 2013

Keywords

Comments

The table is constructed so that row labels are 2^n - 1, and column labels are 2^n. The body of the table is the row*col + 1. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner.
All of these numbers have the following property:
let m be a member of A(n),
if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then
the differences between successive members of B(n) is a repeating series
of 1's with the last difference in the pattern m. The number of ones in
the pattern is 2^j - 1, where j is the column index.
As an example consider A(4) which is 9,
the sequence B(n) where i XOR 8 = i - 8 starts as:
8, 9, 10, 11, 12, 13, 14, 15, 24... (A115419)
with successive differences of:
1, 1, 1, 1, 1, 1, 1, 9.
The main diagonal is the 6th cyclotomic polynomial evaluated at powers of two (A020515).
The formula for diagonals above the main diagonal
2^(2*n+1) - 2^(n + (a+1)/2) + 1 n>=(a+1)/2 a=odd number above diagonal
2^(2*n) - 2^(n + (b/2)) + 1 n>=(b/2)+1 b=even number above diagonal
The formulas for diagonals below the main diagonal
2^(2*n+1) - 2^(n + 1 -(a+1)/2) + 1 n>=(a+1)/2 a=odd number below diagonal
2^(2*n) - 2^(n - (b/2)) + 1 n>=(b/2)+1 b=even number below diagonal
Primes of this sequence are in A152449.

Examples

			Using the lexicographic ordering of A057555 the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
  +1  |    2    4     8    16    32     64    128    256     512    1024 ...
  ----|-----------------------------------------------------------------
  1   |    3    5     9    17    33     65    129    257     513    1025
  3   |    7   13    25    49    97    193    385    769    1537    3073
  7   |   15   29    57   113   225    449    897   1793    3585    7169
  15  |   31   61   121   241   481    961   1921   3841    7681   15361
  31  |   63  125   249   497   993   1985   3969   7937   15873   31745
  63  |  127  253   505  1009  2017   4033   8065  16129   32257   64513
  127 |  255  509  1017  2033  4065   8129  16257  32513   65025  130049
  255 |  511 1021  2041  4081  8161  16321  32641  65281  130561  261121
  511 | 1023 2045  4089  8177 16353  32705  65409 130817  261633  523265
  1023| 2047 4093  8185 16369 32737  65473 130945 261889  523777 1047553
  ...

Links

Crossrefs

Cf. A081118, A152449 (primes), A057555 (lexicographic ordering), A115419 (example).
Rows: A000051(i=1), A181565(2), A083686(3), A195744(4), A206371(5), A196657(6).
Cols: A000225(j=1), A036563(2), A048490(3), A176303 (7 offset of 8).
Diagonals: A020515 (main), A092440, A060867 (above), A134169 (below).

Programs

  • Magma
    //program generates values in a table form
for i:=1 to 10 do
    m:=2^i - 1;
    m,[ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(i-j) + 1;
       if IsPrime(k) then k,"*";
          else k;
       end if;;
    end for;
end for;
  • Mathematica
    Table[(2^j-1)*2^(i-j+1) + 1, {i, 10}, {j, i}] (* Paolo Xausa, Apr 02 2024 *)

Formula

a(n) = (2^(A057555(2*n-1)) - 1)*2^(A057555(2*n)) + 1 for n>=1. [corrected by Jason Yuen, Feb 22 2025]
a(n) = A081118(n)+2; 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

A281482 a(n) = 2^(n + 1) * (2^n + 1) - 1.

Original entry on oeis.org

3, 11, 39, 143, 543, 2111, 8319, 33023, 131583, 525311, 2099199, 8392703, 33562623, 134234111, 536903679, 2147549183, 8590065663, 34360000511, 137439477759, 549756862463, 2199025352703, 8796097216511, 35184380477439, 140737505132543, 562949986975743
Offset: 0

Views

Author

Jaroslav Krizek, Jan 22 2017

Keywords

Links

Crossrefs

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281481 (2^(n - 1) * (2^n + 1) + 1).

Programs

  • Magma
    [2^(n + 1) * (2^n + 1) - 1: n in [0..200]];
  • PARI
    Vec((3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (3 - 10*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

A267816 Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 23, 111, 479, 1983, 8063, 32511, 130559, 523263, 2095103, 8384511, 33546239, 134201343, 536838143, 2147418111, 8589803519, 34359476223, 137438429183, 549754765311, 2199021158399, 8796088827903, 35184363700223, 140737471578111, 562949919866879
Offset: 0

Views

Author

Robert Price, Jan 20 2016

Keywords

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Crossrefs

Cf. A267814.
Similar entries: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A281481 (2^(n - 1) * (2^n + 1) + 1), A281482 (2^(n + 1) * (2^n + 1) - 1). - Jaroslav Krizek, Jan 22 2017

Programs

  • Mathematica
    rule=221; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Formula

Conjectures from Colin Barker, Jan 22 2016 and Apr 16 2019: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-4*x+16*x^2-16*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 2^(n + 1) * (2^n - 1) - 1, for n > 0. - Jaroslav Krizek, Jan 22 2017

A281481 a(n) = 2^(n - 1) * (2^n + 1) + 1.

Original entry on oeis.org

2, 4, 11, 37, 137, 529, 2081, 8257, 32897, 131329, 524801, 2098177, 8390657, 33558529, 134225921, 536887297, 2147516417, 8590000129, 34359869441, 137439215617, 549756338177, 2199024304129, 8796095119361, 35184376283137, 140737496743937, 562949970198529
Offset: 0

Views

Author

Jaroslav Krizek, Jan 22 2017

Keywords

Links

Crossrefs

Similar sequences: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A267816 (2^(n + 1) * (2^n - 1) - 1), A281482 (2^(n + 1) * (2^n + 1) - 1).
Cf. A278930.

Programs

  • Magma
    [2^(n - 1) * (2^n + 1) + 1: n in [0..200]];
  • PARI
    Vec((2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

From Colin Barker, Jan 22 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: (2 - 10*x + 11*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)
a(n) = A278930(n - 2) for n >= 7. - Georg Fischer, Mar 26 2019

A170939 4^n-2^n+2.

Original entry on oeis.org

2, 4, 14, 58, 242, 994, 4034, 16258, 65282, 261634, 1047554, 4192258, 16773122, 67100674, 268419074, 1073709058, 4294901762, 17179738114, 68719214594, 274877382658, 1099510579202, 4398044413954, 17592181850114, 70368735789058, 281474959933442, 1125899873288194
Offset: 0

Views

Author

N. J. A. Sloane, Feb 13 2010

Keywords

Links

Formula

a(n)= 7*a(n-1) -14*a(n-2) +8*a(n-3) = 2*A134169(n). G.f.: 2*(1-5*x+7*x^2)/((1-x) * (2*x-1) * (4*x-1)). [From R. J. Mathar, Feb 15 2010]
Showing 1-6 of 6 results.

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

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