A213859 -id:A213859 - cp's OEIS frontend

A213861 First occurrence of n in A213859.

2, 0, 1, 3, 4, 2949, 8, 11, 12, 15, 17, 115, 20, 7863275, 24, 27, 16, 73, 32, 35, 25, 39, 33, 103, 38, 48589961800007, 228, 51, 119, 97, 56, 59, 47, 323, 52, 581, 69, 71, 43, 2277, 77, 17509, 80, 75, 84, 87, 68, 133, 92, 95, 2209, 99, 53, 29363, 104, 107, 6848, 111, 2585, 3241, 116, 449, 120, 7847, 78, 1111, 129, 173, 132, 135, 137, 5340185

Offset: 0

Alex Ratushnyak, Jun 22 2012

nonn

			Smallest n such that A213859(n) = 7 is 11, so a(7) = 11.

Cf. A213859, A213407, A036236

nn = 25; t = Table[-1, {nn}]; Do[p = PowerMod[2, n, n + 2]; If[0 <= p <= nn && t[[p + 1]] == -1, t[[p + 1]] = n], {n, 0, 10^7}]; t (* T. D. Noe, Jun 26 2012 *)

a(n) = smallest k>n such that 2^k == n (mod k+2).

a(26)-a(50) from T. D. Noe, Jun 26 2012

Terms a(25) and a(51) onward from Max Alekseyev, Feb 01 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

Paul Barry, Apr 28 2005

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A106263 (row sums), A106264 (diagonal sums).

Cf. A000034, A015910, A062173, A112983, A213859, A294389, A294390.

[Modexp(2, n-k, k+2): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 10 2023

Table[PowerMod[2, n-k, k+2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 10 2023 *)

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

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

Alex Ratushnyak, Jul 22 2012

nonn

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

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

A212844 := proc(n)
    modp( 2&^ (n+2),n) ;
end proc: # R. J. Mathar, Jul 24 2012

Table[PowerMod[2, n+2, n], {n, 79}] (* Alonso del Arte, Jul 22 2012 *)

A212844(n)=lift(Mod(2,n)^(n+2)) \\ M. F. Hasler, Jul 23 2012

for n in range(1,99):
    print(2**(n+2) % n, end=',')

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

A294389 a(n) = 2^(n-3) mod n, for n >= 3.

Original entry on oeis.org

1, 2, 4, 2, 2, 0, 1, 8, 3, 8, 10, 4, 1, 0, 13, 8, 5, 12, 1, 6, 6, 8, 4, 20, 10, 16, 22, 8, 8, 0, 1, 26, 11, 8, 28, 10, 1, 32, 31, 8, 11, 24, 19, 12, 12, 32, 16, 28, 1, 28, 40, 44, 26, 32, 1, 44, 15, 32, 46, 16, 1, 0, 4, 8, 17, 36, 1, 58, 18, 8, 55, 56, 46, 40, 60, 8, 20, 32, 10, 62, 21, 8, 4, 22

Offset: 3

Enrique Navarrete, Oct 29 2017

nonn

Every nonnegative integer seems to appear in the sequence, and every integer seems to appear in the sequence of first differences (see link).

For all 3 <= n < 10^9, a(n) != 7. - Robert G. Wilson v, Nov 30 2017

G. C. Greubel, Table of n, a(n) for n = 3..10000
Enrique Navarrete, Sequences Derived from Residues of 2^n (mod n)

Cf. A015910, A212844, A213859.

[Modexp(2,(n-3),n): n in [3..100]]; // G. C. Greubel, Jan 11 2023

Array[PowerMod[2, # - 3, #] &, 80, 3] (* Robert G. Wilson v, Nov 30 2017 *)

[power_mod(2,(n-3),n) for n in range(3,101)] # G. C. Greubel, Jan 11 2023

More terms from Robert G. Wilson v, Nov 30 2017

cp's OEIS Frontend

A213861 First occurrence of n in A213859.

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A106262 An invertible triangle of remainders of 2^n.

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

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A294389 a(n) = 2^(n-3) mod n, for n >= 3.

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