A227990 -id:A227990 - cp's OEIS frontend

A099584 Exponent of 3 in factorization of prime(n) - 1.

0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0, 3, 0, 2, 0, 0, 1, 0, 1, 1, 4, 0, 0, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 3, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 3, 0, 0, 2, 0, 1, 0, 1, 0, 3, 1, 0, 0, 1, 0, 1, 0, 0, 5, 0, 1, 0, 0, 0, 2, 3, 1, 0

Offset: 1

Ralf Stephan, Oct 24 2004

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 1 (mod 3^k) within all the primes is 1/(2*3^(k-1)), for k >= 1. This is also the asymptotic density of terms in this sequence that are >= k. Therefore, the asymptotic density of the occurrences of k in this sequence is d(k) = 1/(2*3^(k-1)) - 1/(2*3^k) = 1/3^k, and the asymptotic mean of this sequence is Sum_{k>=1} k*d(k) = 3/4. - Amiram Eldar, Mar 14 2025

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

Cf. A006093, A007949, A023506, A099585, A227990.

[Valuation(NthPrime(n)-1, 3): n in [1..110]]; // Bruno Berselli, Aug 05 2013

seq(padic:-ordp(ithprime(i)-1,3), i=1..200); # Robert Israel, Mar 02 2018

Table[IntegerExponent[Prime[n] - 1, 3], {n, 110}] (* Bruno Berselli, Aug 05 2013 *)

forprime(p=2, 700, print1(valuation(p-1,3),", ")); \\ Bruno Berselli, Aug 05 2013

a(n) = A007949(A006093(n)).

prime(n) - 1 = 3^a(n) * A099585(n).

A227991 Highest power of 3 dividing prime(n)+1.

Original entry on oeis.org

3, 1, 3, 1, 3, 1, 9, 1, 3, 3, 1, 1, 3, 1, 3, 27, 3, 1, 1, 9, 1, 1, 3, 9, 1, 3, 1, 27, 1, 3, 1, 3, 3, 1, 3, 1, 1, 1, 3, 3, 9, 1, 3, 1, 9, 1, 1, 1, 3, 1, 9, 3, 1, 9, 3, 3, 27, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 3, 9, 1, 1, 1, 3, 3, 1, 3, 1, 3, 1, 27, 1, 1, 3, 9

Offset: 1

Bruno Berselli, Aug 05 2013

nonn

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A068503 (highest power of 3 dividing prime(n)-1), A068504 (highest power of 2 dividing prime(n)+1), A227990.

[3^Valuation(NthPrime(n)+1, 3): n in [1..100]];

Table[3^IntegerExponent[Prime[n] + 1, 3], {n, 100}]

forprime(p=2, 700, print1(3^valuation(p+1, 3), ", "));

a(n) = 3^A227990(n).

cp's OEIS Frontend

A099584 Exponent of 3 in factorization of prime(n) - 1.

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