A283419 - OEIS

%I #52 Feb 16 2025 08:33:42

%S 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,

%T 34,69,36,73,38,85,44,46,93,50,101,105,109,56,58,60,121,64,129,66,133,

%U 141,149,76,153,78,80,161,84,86,88,90,181,185,94,189,98,205,104,209,106,221,225

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

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/ModularInverse.html">Modular Inverse</a>

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

%F 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

%F a(n) = ceiling(2p/3)/(p mod 3) where p is the n-th prime, a(2)=0. - _Travis Scott_, Feb 08 2023

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

%p a:= n-> `if`(n=2, 0, (p-> ceil(2*p/3)/(p mod 3))(ithprime(n))):

%p seq(a(n), n=1..75);  # _Alois P. Heinz_, Feb 08 2023

%t a[n_] := ModularInverse[3, Prime[n]]; Table[a[n], {n, 3, 100}]

%t Table[If[Mod[Prime@n,3]==0,0,ModularInverse[3,Prime@n]],{n,88}]

%o (PARI) a(n) = if (n==2, 0, lift(1/Mod(3, prime(n)))); \\ _Michel Marcus_, Mar 31 2021

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

%K nonn,changed

%O 1,3

%A _Jean-François Alcover_, May 02 2017

%E Name edited and a(1)-a(2) prepended by _Travis Scott_, Feb 08 2023