A385073 a(n) = b^(n-1) mod n, where b = A053669(n) is the least integer greater than 1 and coprime to n.

0, 1, 1, 3, 1, 5, 1, 3, 4, 3, 1, 5, 1, 3, 4, 11, 1, 11, 1, 7, 4, 3, 1, 5, 16, 3, 13, 27, 1, 7, 1, 11, 4, 3, 9, 29, 1, 3, 4, 27, 1, 17, 1, 27, 31, 3, 1, 29, 15, 33, 4, 27, 1, 11, 49, 3, 4, 3, 1, 43, 1, 3, 4, 43, 16, 23, 1, 27, 4, 13, 1, 29, 1, 3, 34, 27, 9, 5, 1, 27, 40, 3, 1, 17

Offset

1,4

Comments

Inspired by Fermat's Little Theorem.

a(n) > 0 for n > 1 since n and b are coprime.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Fermat's little theorem

Formula

a(n) = 0 iff n = 1.

a(n) = 1 iff n belongs to A000040, A001567, or A130433.

a(n) = 2 iff n>1 and belongs to A173572;

a(n) = 4 iff n belongs to A033553;

a(n) = 8 iff n>7 and belongs to either A033984 or A173138;

a(n) = 16 iff n>15 and belongs to A276968;

a(n) = 32 iff n>1 and belongs to A215610;

a(n) = 64 iff n>63 and belongs to A276969;

a(n) = 128 iff n>127 and belongs to A215611;

a(n) = 256 iff n>255 and belongs to A276970;

a(n) = 512 iff n>511 and belongs to A215612;

a(n) = 1024 iff n>1023 and belongs to A276971;

a(n) = 2048 iff n>2047 and belongs to A215613;

From Robert Israel, Jun 18 2025: (Start)

a(2*p) = 3 if p is a prime > 3.

a(3*p) = 4 if p is a prime > 2.

a(4*p) = 3^3 if p is a prime > 5.

a(6*p) = 5^5 if p is a prime > 509.

a(8*p) = 3^5 if p is a prime > 271.

a(10*p) = 3^9 if p is a prime > 1951.

a(12*p) = 5^11 if p is a prime > 4069003. (End)

Maple

f:= proc(n) local b;
  b:= 2;
  while n mod b = 0 do b:= nextprime(b) od;
  b &^ (n-1) mod n
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Jun 18 2025

Mathematica

a[n_] := Block[{b = 2}, While[GCD[n, b] > 1, b++]; PowerMod[b, n - 1, n]]; Array[a, 84]

Prog

(PARI) a(n) = forprime(p=2, , if(n%p, return(lift(Mod(p, n)^(n-1))))); \\ Michel Marcus, Jun 18 2025

Crossrefs

Cf. A053669, A385074.

Sequence in context: A385047 A396262 A387975 * A243158 A154723 A273262

Adjacent sequences: A385070 A385071 A385072 * A385074 A385075 A385076

Keyword

nonn,look

Author

Robert G. Wilson v, Jun 16 2025

Status

approved