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A067775 Primes p such that p + 4 is composite. 6
2, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 359, 367, 373, 383, 389 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes n such that n!*B(n+3) is an integer where B(k) are the Bernoulli numbers B(1) = -1/2, B(2) = 1/6, B(4) = -1/30, ..., B(2m+1) = 0 for m > 1.

If n is prime n!*B(n-1) is always an integer. Note that if Goldbach's conjecture (2n = p1 + p2 for all n >= 2) is false and K is the smallest value of n for which it fails, then for 2(K-2) = p3 + p4, the primes p3 and p4 must be taken from this list. See similar comment for A140555. - Keith Backman, Apr 06 2012

Complement of A023200 (primes p such that p + 4 is also prime) with respect to A000040 (primes). For n > 1: a(n) = primes p such that there is no prime of the form r^2 + p where r is prime, a(n) = subsequence of A232010. Example: the prime 7 is not in sequence because 2^2 + 7 = 11 (prime). A232009(a(n)) = 0 for n > 1 . - Jaroslav Krizek, Nov 22 2013

LINKS

Klaus Brockhaus, Table of n, a(n) for n = 1..1026

Eric Weisstein's World of Mathematics, Bernoulli Number

FORMULA

a(n) ~ n log n. - Charles R Greathouse IV, Nov 22 2013

MATHEMATICA

A067775 = {}; Do[p = Prime@ n; If[ IntegerQ[ p! BernoulliB[p + 3]], AppendTo[A067775, p]], {n, 77}]; A067775 (* Robert G. Wilson v, Aug 19 2008 *)

Select[Prime[Range[80]], Not[PrimeQ[# + 4]] &] (* Alonso del Arte, Apr 02 2014 *)

PROG

(PARI) lista(nn) = {forprime(p=1, nn, if (! isprime(p+4), print1(p, ", ")); ); } \\ Michel Marcus, Nov 22 2013

CROSSREFS

Cf. A049591, A140555, A232009, A232010, A232012, A023200.

Sequence in context: A181580 A118754 A140559 * A140552 A138644 A164921

Adjacent sequences:  A067772 A067773 A067774 * A067776 A067777 A067778

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Feb 06 2002

EXTENSIONS

New name from Klaus Brockhaus on the suggestion of Michel Marcus, Nov 22 2013

STATUS

approved

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Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)