A043297 Primes p such that B(4*p) has denominator 30 where B(2n) are the Bernoulli numbers.

2, 17, 19, 31, 47, 59, 61, 71, 101, 103, 107, 109, 137, 149, 151, 157, 167, 181, 197, 211, 223, 227, 229, 241, 257, 263, 269, 271, 283, 311, 313, 317, 331, 337, 347, 349, 353, 367, 379, 383, 389, 397, 401, 421, 439, 449, 457, 461, 463, 467, 479, 503, 521

Offset

1,1

Comments

Complement of A087634, primes p such that phi(k) = 4p has a solution, where phi is Euler's totient function.

The sequences a(n), A005384 and A023212 form a partition of the set of primes > 3: Using von Staudt-Clausen formula the divisors of 4p increased by 1 are {2,3,5,p+1,2p+1,4p+1}, p+1 is clearly an even number, and if 2p+1 and 4p+1 are not prime, then denom(B(4p))=30. - Enrique Pérez Herrero, Aug 15 2011

Also 2 with the primes p such that both 2*p+1 and 4*p+1 are composite: A210684. For these numbers k > 2 the equation: phi(n)=k*tau(n), where phi is A000010 and tau is A000005, has no solutions: A112954(a(n))=0. - Enrique Pérez Herrero, May 12 2012

Links

E. Pérez Herrero, Table of n, a(n) for n=1..30000

Wikipedia, Von Staudt-Clausen theorem

Mathematica

Select[Prime[Range[100]], Denominator[BernoulliB[4# ]]==30&] (* T. D. Noe, Feb 19 2004 *)
Select[Prime[Range[100]], !PrimeQ[4#+1]&&!PrimeQ[2#+1]||(#==2)&] (* Enrique Pérez Herrero, Aug 16 2011 *)

Crossrefs

Cf. A051225, A053176, A087634.

Cf. A005384, A023212.

Cf. A210684.

Sequence in context: A153261 A080115 A119480 * A018569 A022118 A041339

Adjacent sequences: A043294 A043295 A043296 * A043298 A043299 A043300

Keyword

easy,nonn

Author

Benoit Cloitre, Mar 24 2002

Status

approved