A099584 -id:A099584 - cp's OEIS frontend

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

Offset: 1

Bruno Berselli, Aug 05 2013

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

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

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

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

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

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

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

A068503 Highest power of 3 dividing prime(n)-1.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 9, 1, 1, 3, 9, 1, 3, 1, 1, 1, 3, 3, 1, 9, 3, 1, 1, 3, 1, 3, 1, 27, 1, 9, 1, 1, 3, 1, 3, 3, 81, 1, 1, 1, 9, 1, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 27, 3, 1, 3, 1, 9, 1, 3, 1, 3, 3, 1, 3, 1, 1, 3, 3, 27, 1, 1, 9, 1, 3, 1, 3, 1, 27, 3, 1, 1, 3, 1, 3, 1, 1, 243, 1, 3, 1, 1, 1, 9, 27

Offset: 1

Benoit Cloitre, Mar 11 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006093, A038500, A099584.

a[n_] := 3^IntegerExponent[Prime[n] - 1, 3]; Array[a, 100] (* Amiram Eldar, May 14 2022 *)

a(n) = 3^valuation(prime(n)-1, 3); \\ Michel Marcus, Nov 24 2013

a(n) = A038500(A006093(n)). - Michel Marcus, Nov 24 2013

a(n) = 3^A099584(n). - Michel Marcus, Nov 24 2013

A099585 Remove all 3s from prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 2, 10, 4, 16, 2, 22, 28, 10, 4, 40, 14, 46, 52, 58, 20, 22, 70, 8, 26, 82, 88, 32, 100, 34, 106, 4, 112, 14, 130, 136, 46, 148, 50, 52, 2, 166, 172, 178, 20, 190, 64, 196, 22, 70, 74, 226, 76, 232, 238, 80, 250, 256, 262, 268, 10, 92, 280, 94, 292, 34, 310

Offset: 1

Ralf Stephan, Oct 24 2004

nonn

Equals A038502( A006093(n) ). prime(n)-1 = a(n) * 3^A099584(n).

Cf. A057023.

a(n) = (prime(n)-1)/3^valuation(prime(n)-1,3)

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A227990 3^a(n) is the highest power of 3 dividing prime(n)+1.

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A068503 Highest power of 3 dividing prime(n)-1.

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A099585 Remove all 3s from prime(n) - 1.

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