A283419 a(n) is the multiplicative inverse of 3 modulo the n-th prime (or 0 if none exists).

1, 0, 2, 5, 4, 9, 6, 13, 8, 10, 21, 25, 14, 29, 16, 18, 20, 41, 45, 24, 49, 53, 28, 30, 65, 34, 69, 36, 73, 38, 85, 44, 46, 93, 50, 101, 105, 109, 56, 58, 60, 121, 64, 129, 66, 133, 141, 149, 76, 153, 78, 80, 161, 84, 86, 88, 90, 181, 185, 94, 189, 98, 205, 104, 209, 106, 221, 225

Offset: 1

Author

Jean-François Alcover, May 02 2017

Keywords

• nonn

Examples

			3*5 mod prime(4) = 15 mod 7 = 1, so a(4) = 5.
		

Links

• Eric Weisstein's MathWorld, Modular Inverse

Crossrefs

Cf. A006254 (modular inverses of 2 modulo the odd primes).

Programs

Maple

a:= n-> `if`(n=2, 0, (p-> ceil(2*p/3)/(p mod 3))(ithprime(n))):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 08 2023

Mathematica

a[n_] := ModularInverse[3, Prime[n]]; Table[a[n], {n, 3, 100}]
Table[If[Mod[Prime@n,3]==0,0,ModularInverse[3,Prime@n]],{n,88}]

PARI

a(n) = if (n==2, 0, lift(1/Mod(3, prime(n)))); \\ Michel Marcus, Mar 31 2021

Formula

3 * a(n) == 1 (mod prime(n)).

a(n) = (p * (-1)^((r+1)/2))/3 mod p = q * (-1)^((r+1)/2) mod p where p = prime(n) = 3*q + r, with r = -1 or 1 and quotient q a positive integer. - Ian George Walker, Mar 24 2021

a(n) = ceiling(2p/3)/(p mod 3) where p is the n-th prime, a(2)=0. - Travis Scott, Feb 08 2023

Extensions

Name edited and a(1)-a(2) prepended by Travis Scott, Feb 08 2023