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 1-4 of 4 results.

A175347 Primes in A176303.

Original entry on oeis.org

140737488355201, 36028797018963841, 10141204801825835211973625642881, 11150372599265311570767859136324180752990081
Offset: 1

Views

Author

Zak Seidov, Apr 18 2010

Keywords

Comments

The next term is too large to be displayed here. To extend the sequence, use A169716 to obtain the exponents n.
a(1)..a(6) are 2^n-127 for n = 47, 55, 103, 143, 391, 2807; see A169716.

Crossrefs

Cf. A176303, A169716.

Formula

a(n) = 2^A169716(n)-127.

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

A169716 Numbers k such that 2^k - 127 is prime.

Original entry on oeis.org

47, 55, 103, 143, 391, 2807, 11647, 19223, 264655
Offset: 1

Views

Author

Zak Seidov, Apr 18 2010

Keywords

Comments

Exponents k arising in A175347: see that sequence for the actual primes.
The next term is > 10^4. [M. F. Hasler, Apr 18 2010]
All terms are == 7 (mod 8). [M. F. Hasler, Apr 18 2010]
No more terms through 78815. - Charles R Greathouse IV, Apr 19 2010
a(10) > 300000. - Michael S. Branicky, Jul 11 2025

Links

Crossrefs

Cf. A175347, A176303.

Programs

Extensions

More terms from M. F. Hasler, Apr 18 2010
a(7)-a(8) from Charles R Greathouse IV, Apr 19 2010
a(9) from Michael S. Branicky, Jul 10 2025

A176494 Least m >= 1 for which |2^m - prime(n)| is prime.

Original entry on oeis.org

3, 1, 1, 2, 1, 2, 1, 2, 4, 1, 3, 2, 1, 2, 4, 4, 1, 3, 2, 1, 3, 2, 4, 3, 2, 1, 2, 1, 2, 47, 2, 6, 1, 8, 1, 3, 5, 2, 4, 4, 1, 6, 1, 2, 1, 5, 5, 2, 1, 2, 4, 1, 8, 4, 6, 8, 1, 3, 2, 1, 4, 7, 2, 1, 2, 9, 791, 4, 1, 2, 8, 3, 9, 5, 2, 4, 3, 2, 3, 8, 1, 6, 1, 3, 2, 4, 3, 2, 1, 2, 4, 3, 2, 3
Offset: 2

Views

Author

Vladimir Shevelev, Apr 19 2010, Aug 15 2010

Keywords

Comments

a(n)=1 iff p_n is second of twin primes (A006512); for n > 4, a(n)=2 iff p_n is second of cousin primes (A046132). It is interesting to continue this sequence in order to find big jumps such as a(31)-a(30). Is it true that such jumps can be arbitrarily large, either (a) in the sense of differences a(n+1)-a(n), or (b) in the sense of ratios a(n+1)/a(n)?
Conjecture. For every odd prime p, the sequence {|2^n - p|} contains at least one prime. The record values of the sequence appear at n = 2, 10, 31, 68, 341, ... and are 3, 4, 47, 791, ... Note that up to now the value a(341) is not known. Charles R Greathouse IV calculated the following two values: a(815)=16464, a(591)=58091 and noted that a(341) is much larger [private communication, May 27 2010]. - Vladimir Shevelev, May 29 2010

Links

Crossrefs

Cf. A176303, A175347, A006512, A046132.

Programs

  • Mathematica
    lm[n_]:=Module[{m=1},While[!PrimeQ[Abs[2^m-n]],m++];m]; Table[lm[i],{i,Prime[ Range[2,100]]}] (* Harvey P. Dale, Aug 11 2014 *)

Extensions

Beginning with a(31), the terms were calculated by Zak Seidov - private communication, Apr 20 2010
Sequence extended by R. J. Mathar via the Seqfan Discussion List, Aug 15 2010
Showing 1-4 of 4 results.

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