A073409 -id:A073409 - cp's OEIS frontend

A385958 a(n) is the largest prime p such that b(n) = b(n-1)*(p+1)/(p-1) is an integer (A385959), where b(0) = 1.

3, 5, 7, 5, 13, 3, 29, 31, 17, 37, 3, 5, 7, 5, 229, 47, 241, 23, 89, 271, 137, 277, 3, 557, 19, 311, 313, 5, 7, 5, 13, 3, 4397, 7, 5, 13, 3, 29, 21991, 5, 13, 3, 29, 82471, 677, 733, 227, 27893, 19, 11, 111577, 3, 5, 283, 5, 505663, 15803

Offset: 1

Thomas Ordowski, Jul 13 2025

a(n) = (b(n)+b(n-1))/(b(n)-b(n-1)), where b(n) = A385959(n) is the smallest k such that a(n) is a prime, where b(0) = 1.
a(n) is the largest prime p such that p-1 divides 2*b(n-1).
Note that 3 <= a(n) <= 2*b(n-1)+1.
Does this sequence contain all odd primes?

Martin Fuller, Table of n, a(n) for n = 1..3460

Cf. A065091, A073409, A385959.

allocatemem(2^30);
default(factor_add_primes, 1);
{
my(a,b=1);
for(n=1,100,
  removeprimes(select(p->b%p, addprimes()));
  fordiv(2*b, d, a=2*b/d+1; if(isprime(a),break));
  b+=b*2/(a-1);
  print1(a, ", ");
);
} \\ Martin Fuller, Jul 16 2025

a(n) = A073409(b(n-1)), where b(n) = A385959(n) = Product_{k=1..n} (a(k)+1)/(a(k)-1).

Also tanh(Sum_{k=1..n} arctanh(1/a(k))) = (b(n)-1)/(b(n)+1).

More terms from Morné Louw and Martin Fuller, Jul 15 2025

A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 5, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 11, 1, 23, 1, 13, 1, 3, 1, 29, 1, 31, 1, 17, 1, 3, 1, 37, 1, 3, 1, 41, 1, 43, 1, 23, 1, 47, 1, 17, 1, 11, 1, 53, 1, 19, 1, 29, 1, 59, 1, 61, 1, 3, 1, 17, 1, 67, 1, 5, 1, 71, 1, 73, 1, 3, 1, 5, 1, 79, 1, 41, 1, 83, 1, 43, 1, 3, 1, 89, 1

Offset: 0

Paul Curtz, Oct 12 2009

nonn

The largest member of the extended prime list A008578 which divides the denominator of Bernoulli(n).

Essentially A073409 padded with 1's.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A006530, A027642, A073409, A089026, A166120.

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A166333(n) = A006530(denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(n) = A006530(A027642(n)). - Antti Karttunen, Dec 19 2018

Edited and extended by R. J. Mathar, Oct 21 2009

Name and comment swapped by Antti Karttunen, Dec 19 2018

cp's OEIS Frontend

A385958 a(n) is the largest prime p such that b(n) = b(n-1)*(p+1)/(p-1) is an integer (A385959), where b(0) = 1.

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