A036134 a(n) = 3^n mod 79.

1, 3, 9, 27, 2, 6, 18, 54, 4, 12, 36, 29, 8, 24, 72, 58, 16, 48, 65, 37, 32, 17, 51, 74, 64, 34, 23, 69, 49, 68, 46, 59, 19, 57, 13, 39, 38, 35, 26, 78, 76, 70, 52, 77, 73, 61, 25, 75, 67, 43, 50, 71, 55, 7, 21, 63, 31, 14

Offset

0,2

Comments

Because a(39) = 78, the Legendre symbol (3/79) = -1, meaning that 3 is not a quadratic residue of 79. Furthermore, it means that 3 is prime in Z[sqrt(79)]. - Alonso del Arte, Oct 01 2012

References

I. M. Vinogradov, Elements of Number Theory, pp. 220 ff.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1).

Formula

From G. C. Greubel, Oct 17 2018: (Start)

a(n) = a(n-1) - a(n-39) + a(n-40).

a(n+78) = a(n). (End)

Example

a(4) = 2 because 3^4 = 81 and 81 - 79 = 2.

Maple

[ seq(primroot(ithprime(i))^j mod ithprime(i), j=0..100) ];

Mathematica

Table[Mod[3^n, 79], {n, 0, 60}] (* Alonso del Arte, Oct 01 2012 *)
PowerMod[3, Range[0, 100], 79] (* Harvey P. Dale, Feb 21 2024 *)

Prog

(PARI) a(n)=lift(Mod(3, 79)^n) \\ Charles R Greathouse IV, Mar 22 2016
(Magma) [Modexp(3, n, 79): n in [0..100]]; // G. C. Greubel, Oct 17 2018
(Python) for n in range(0, 100): print(int(pow(3, n, 79)), end=' ') # Stefano Spezia, Oct 17 2018
(GAP) List([0..60], n->PowerMod(3, n, 79)); # Muniru A Asiru, Oct 17 2018

Crossrefs

Cf. A000244 (3^n).

Sequence in context: A126025 A317497 A114181 * A317502 A213912 A070360

Adjacent sequences: A036131 A036132 A036133 * A036135 A036136 A036137

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved