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.

A187787 Numbers k such that 2^(k+1) == 1 (mod k).

Original entry on oeis.org

1, 3, 15, 35, 119, 255, 455, 1295, 2555, 2703, 3815, 3855, 4355, 5543, 6479, 8007, 9215, 10439, 10619, 11951, 16211, 22895, 23435, 26319, 26795, 27839, 28679, 35207, 43055, 44099, 47519, 47879, 49679, 51119, 57239, 61919, 62567, 63167, 63935, 65535, 74447, 79055
Offset: 1

Views

Author

Franz Vrabec, Jan 06 2013

Keywords

Comments

Prime factorizations of the first ten terms: 3, 3*5, 5*7, 7*17, 3*5*17, 5*7*13, 5*7*37, 5*7*73, 3*17*53, 5*7*109.

Examples

			3 is in the sequence because 2^(3+1) mod 3 = 16 mod 3 = 1.

Links

Crossrefs

Cf. A055685, A069927, A112983.

Programs

  • Maple
    for n from 1 to 100000 do if 2&^(n+1) mod n = 1 then print(n) fi od;
  • Mathematica
    m = 1; Join[Select[Range[1, m], Divisible[2^(# + 1), #] &],
Select[Range[m + 1, 10^5], PowerMod[2, # + 1, #] == m &]] (* Robert Price, Oct 11 2018 *)
Join[{1},Select[Range[80000],PowerMod[2,#+1,#]==1&]] (* Harvey P. Dale, Aug 19 2019 *)
  • PARI
    for (n=1,10^7, if (Mod(2,n)^(n+1)==1,print1(n,", "))); /* Joerg Arndt, Jan 06 2013 */

Extensions

Term a(1)=1 prepended by Max Alekseyev, Nov 29 2014

A106262 An invertible triangle of remainders of 2^n.

Original entry on oeis.org

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

Views

Author

Paul Barry, Apr 28 2005

Keywords

Examples

			Triangle begins:
  1;
  0, 1;
  0, 2, 1;
  0, 1, 2, 1;
  0, 2, 0, 2, 1;
  0, 1, 0, 4, 2, 1;
  0, 2, 0, 3, 4, 2, 1;
  0, 1, 0, 1, 2, 4, 2, 1;
  0, 2, 0, 2, 4, 1, 4, 2, 1;
  0, 1, 0, 4, 2, 2, 0, 4, 2, 1;

Links

Crossrefs

Cf. A106263 (row sums), A106264 (diagonal sums).
Cf. A000034, A015910, A062173, A112983, A213859, A294389, A294390.

Programs

  • Magma
    [Modexp(2, n-k, k+2): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 10 2023
  • Mathematica
    Table[PowerMod[2, n-k, k+2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 10 2023 *)
  • SageMath
    flatten([[power_mod(2,n-k,k+2) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 10 2023

Formula

T(n, k) = 2^(n-k) mod (k+2).
Sum_{k=0..n} T(n, k) = A106263(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A106264(n) (diagonal sums).
From G. C. Greubel, Jan 10 2023: (Start)
T(n, 0) = A000007(n).
T(n, 1) = A000034(n+1).
T(2*n, n) = A213859(n).
T(2*n, n-1) = A015910(n+1).
T(2*n, n+1) = A294390(n+3).
T(2*n+1, n-1) = A112983(n+1).
T(2*n+1, n+1) = A294389(n+3).
T(2*n-1, n-1) = A062173(n+1). (End)

A212844 a(n) = 2^(n+2) mod n.

Original entry on oeis.org

0, 0, 2, 0, 3, 4, 1, 0, 5, 6, 8, 4, 8, 2, 2, 0, 8, 4, 8, 4, 11, 16, 8, 16, 3, 16, 23, 8, 8, 16, 8, 0, 32, 16, 2, 4, 8, 16, 32, 24, 8, 4, 8, 20, 23, 16, 8, 16, 22, 46, 32, 12, 8, 4, 7, 16, 32, 16, 8, 4, 8, 16, 32, 0, 63, 58, 8, 64, 32, 36, 8, 40, 8, 16, 47
Offset: 1

Views

Author

Alex Ratushnyak, Jul 22 2012

Keywords

Comments

Also a(n) = x^x mod (x-2), where x = n+2.
Indices of 0's: 2^k, k>=0.
Indices of 1's: 7, 511, 713, 11023, 15553, 43873, 81079, 95263, 323593, 628153, 2275183, 6520633, 6955513, 7947583, 10817233, 12627943, 14223823, 15346303, 19852423, 27923663, 28529473, ...
Conjecture: every integer k >= 0 appears in a(n) at least once.
Each number below 69 appears at least once. Some large first occurrences: a(39806401) = 25, a(259274569) = 33, a(10571927) = 55, a(18039353) = 81. - Charles R Greathouse IV, Jul 21 2015

Examples

			a(3) = 2^5 mod 3 = 32 mod 3 = 2.

Links

Crossrefs

Cf. A000312, A015910, A062173, A112983, A213381, A213859.

Programs

  • Maple
    A212844 := proc(n)
    modp( 2&^ (n+2),n) ;
end proc: # R. J. Mathar, Jul 24 2012
  • Mathematica
    Table[PowerMod[2, n+2, n], {n, 79}] (* Alonso del Arte, Jul 22 2012 *)
  • PARI
    A212844(n)=lift(Mod(2,n)^(n+2)) \\ M. F. Hasler, Jul 23 2012
  • Python
    for n in range(1,99):
    print(2**(n+2) % n, end=',')

Formula

a(n) = 2^(n+2) mod n.

A112985 2^(2^n mod n-1).

Original entry on oeis.org

1, 4, 1, 1, 2, 1, 16, 4, 16, 1, 128, 256, 16, 256, 16, 256, 2, 1, 16, 4, 16, 4096, 65536, 256, 16, 256, 16384, 256, 33554432, 16, 16, 256, 16, 1, 65536, 256, 2, 1048576, 16, 256, 65536, 4294967296, 16, 4, 16, 4294967296, 17179869184, 256, 16, 4294967296, 2048
Offset: 0

Views

Author

Paul Barry, Oct 08 2005

Keywords

Links

Programs

  • Mathematica
    Join[{1,4},Table[2^PowerMod[2,n,n-1],{n,2,60}]] (* Harvey P. Dale, Jan 17 2017 *)

Formula

a(n)=2^A112983(n).

A281375 a(n) = floor(2^(n+1)/n).

Original entry on oeis.org

4, 4, 5, 8, 12, 21, 36, 64, 113, 204, 372, 682, 1260, 2340, 4369, 8192, 15420, 29127, 55188, 104857, 199728, 381300, 729444, 1398101, 2684354, 5162220, 9942053, 19173961, 37025580, 71582788, 138547332, 268435456, 520602096, 1010580540, 1963413621, 3817748707, 7429132620, 14467258260, 28192605840, 54975581388
Offset: 1

Views

Author

Robert Israel, Jan 20 2017

Keywords

Links

Crossrefs

Cf. A053639, A112983, A191636.

Programs

Formula

a(n) = A053639(n) if n is in A000079, otherwise A053639(n)-1.
a(2k-1) = A191636(k) for k > 1.
a(n) = (2^(n+1)-A112983(n))/n.

A112984 Numbers k such that 2^k mod k-1 is odd.

Original entry on oeis.org

4, 10, 16, 28, 36, 50, 56, 76, 82, 106, 116, 120, 122, 134, 144, 162, 172, 176, 188, 190, 204, 216, 218, 232, 236, 244, 248, 254, 256, 262, 274, 280, 290, 296, 298, 300, 324, 334, 336, 344, 352, 364, 372, 396, 404, 406, 414, 426, 438, 452, 456, 474, 476, 484
Offset: 1

Views

Author

Paul Barry, Oct 08 2005

Keywords

Crossrefs

Cf. A112983, A015911.

Programs

  • Mathematica
    Select[ 2Range[250], OddQ[ PowerMod[2, #, #-1]] &] (* Robert G. Wilson v, Oct 10 2005 *)

Extensions

More terms from Robert G. Wilson v, Oct 10 2005
Showing 1-6 of 6 results.

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