A318623 a(n) = 2^phi(n) mod n.

0, 0, 1, 0, 1, 4, 1, 0, 1, 6, 1, 4, 1, 8, 1, 0, 1, 10, 1, 16, 1, 12, 1, 16, 1, 14, 1, 8, 1, 16, 1, 0, 1, 18, 1, 28, 1, 20, 1, 16, 1, 22, 1, 12, 1, 24, 1, 16, 1, 26, 1, 40, 1, 28, 1, 8, 1, 30, 1, 16, 1, 32, 1, 0, 1, 34, 1, 52, 1, 36, 1, 64, 1, 38, 1, 20, 1, 40, 1

Offset

1,6

Comments

Of course, a(n) = 0 iff n is a power of 2 and a(n) = 1 iff n is an odd number > 1. For other n, let n = 2^t*s, t > 0, s > 1 is an odd number, then a(n) is the unique solution to x == 0 (mod 2^t) and x == 1 (mod s).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

If n is a power of 2 then a(n) = 0; if n is an odd number > 1 then a(n) = 1; else, let n = 2^t*s, t > 0, s > 1 is an odd number, then a(n) = n - (s mod 2^t)^2 + 1.

Example

a(6) = 2^phi(6) mod 6 = 2^4 mod 6 = 4.
a(18) = 2^phi(18) mod 18 = 2^6 mod 18 = 10.

Mathematica

a[n_] = Mod[2^EulerPhi[n], n]; Array[a, 50] (* Stefano Spezia, Sep 01 2018 *)
Table[PowerMod[2, EulerPhi[n], n], {n, 80}] (* Harvey P. Dale, Nov 07 2021 *)

Prog

(PARI) a(n) = lift(Mod(2, n)^(eulerphi(n)))
(Magma) [Modexp(2, EulerPhi(n), n): n in [1..110]]; // Vincenzo Librandi, Aug 02 2018

Crossrefs

Cf. A000010, A007663, A245970.

Sequence in context: A290456 A010639 A035588 * A332055 A073027 A370073

Adjacent sequences: A318620 A318621 A318622 * A318624 A318625 A318626

Keyword

nonn,easy

Author

Jianing Song, Aug 30 2018

Status

approved