A294389 -id:A294389 - cp's OEIS frontend

A106262 An invertible triangle of remainders of 2^n.

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)

A294390 a(n) = 2^(n-4) mod n, for n >= 4.

Original entry on oeis.org

1, 2, 4, 1, 0, 5, 4, 7, 4, 5, 2, 8, 0, 15, 4, 12, 16, 11, 14, 3, 16, 2, 10, 5, 8, 11, 4, 4, 0, 17, 30, 23, 4, 14, 24, 20, 16, 36, 4, 27, 12, 32, 6, 6, 16, 8, 14, 26, 40, 20, 22, 13, 16, 29, 22, 37, 16, 23, 8, 32, 0, 2, 4, 42, 52, 35, 64, 9, 40, 64, 28, 23, 20, 30, 4

Offset: 4

Enrique Navarrete, Oct 29 2017

Every nonnegative integer seems to appear in the sequence, and every integer seems to appear in the sequence of first differences (see link).
From Robert Israel, Dec 04 2017: (Start)
a(n) = 0 iff n>=8 is a power of 2.
a(n) = 1 iff n=4 or n is in A033984.
a(n) = 2 iff n>=4 is in A015925 and is not divisible by 4. (End)

			For n=9, 2^5 = 32 == 5 mod 9.

Robert Israel, Table of n, a(n) for n = 4..10000
Enrique Navarrete, Sequences derived from residues of 2^n (mod n)

Cf. A015910, A015925, A033984, A294389.

[Modexp(2, n-4, n): n in [4..120]]; // G. C. Greubel, Dec 27 2024

A294390:=n->2&^(n-4) mod n: seq(A294390(n), n=4..150); # Wesley Ivan Hurt, Nov 30 2017

Array[Mod[2^(# - 4), #] &, 75, 4] (* Michael De Vlieger, Dec 02 2017 *)
Array[PowerMod[2,#-4,#]&,80,4] (* Harvey P. Dale, Dec 01 2018 *)

a(n) = lift(Mod(2, n)^(n-4)); \\ Michel Marcus, Oct 30 2017

print([power_mod(2, n-4, n) for n in range(4, 101)]) # G. C. Greubel, Dec 27 2024

More terms from Michel Marcus, Oct 30 2017

cp's OEIS Frontend

A106262 An invertible triangle of remainders of 2^n.

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