Voozh

A227990

3^a(n) is the highest power of 3 dividing prime(n)+1.

1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 1, 3, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 1, 3, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 2

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OFFSET

1,7

COMMENTS

This is the 3-adic valuation of prime(n)+1.

By Dirichlet's theorem on arithmetic progressions, the asymptotic density of primes p such that p == 3^k-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

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Bruno Berselli)

FORMULA

a(n) = A007949(A008864(n)).

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A007949, A008864, A023512 (2-adic valuation of prime(n)+1), A099584 (3-adic valuation of prime(n)-1), A227991 (associated powers of 3).

Sequence in context: A156348 A306437 A343746 * A101614 A051659 A085861