

A092307


Primes p such that there are no primes q, 3 < q < p, such that (q1) divides (p1).


11



5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 239, 263, 347, 359, 383, 443, 467, 479, 503, 563, 587, 647, 659, 719, 827, 839, 863, 887, 983, 1019, 1187, 1223, 1259, 1283, 1307, 1319, 1367, 1439, 1487, 1499, 1523, 1619, 1787, 1823, 1847, 1907, 2027, 2039, 2063
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OFFSET

1,1


COMMENTS

Using a sieve, these primes can be generated quickly. In the set of primes < 10^9, the density of these primes is about 1/10. It is easy to show that this sequence contains all "safe" primes (A005385).
Primes p such that 6p is the denominator of some Bernoulli number.  T. D. Noe, Sep 26 2006
Except for 5 and 7, primes p such that 12p is the denominator of B(p  1)/(p  1) where B(n) is the Bernoulli number. [Peter Luschny, Dec 24 2008]


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
P. Luschny, Von Staudt prime number, definition and computation. [From Peter Luschny, Dec 24 2008]


FORMULA

Let h(x) = 12x(x + log(exp(x) 1)  log(x)) and [x^n]S(h) denote the coefficient of x^n in the series expansion of h. Consider for n > 1 the relation n = denominator((n  1)![x^n]S(h)). [Peter Luschny, Dec 24 2008]


EXAMPLE

11 is in the sequence because 10 is not a multiple of either 4 or 6.
13 is not in the sequence because, although 12 is not a multiple of 6 or 10, it is a multiple of 4.


MAPLE

For p>7: seq(`if`(denom(bernoulli(n1)/(n1))=12*n, n, NULL), n=2..500); # Peter Luschny, Dec 24 2008


MATHEMATICA

t = Table[p = Prime[n]; Length[Select[Divisors[p  1] + 1, PrimeQ]], {n, 311}]; Prime[Flatten[Position[t, 3]]]


PROG

(Perl) use ntheory ":all"; forprimes { say if (bernfrac($_1))[1] == 6*$_ } 1000; # Dana Jacobsen, Dec 29 2015
(Perl) use ntheory ":all"; forprimes { my $p=$_; say if vecnone { $_ > 3 && $_ < $p1 && is_prime($_+1) } divisors($p1); } 5, 1000; # Dana Jacobsen, Dec 29 2015


CROSSREFS

Cf. A090801 (distinct numbers appearing as denominators of Bernoulli numbers)
Cf. A092308 (for p=prime(n), the number of primes q such that q1 divides p1).
Cf. A005385 (primes p such that (p1)/2 is also prime).
Cf. A152951. [From Peter Luschny, Dec 24 2008]
Sequence in context: A151715 A226027 A090810 * A005385 A181602 A075705
Adjacent sequences: A092304 A092305 A092306 * A092308 A092309 A092310


KEYWORD

nonn


AUTHOR

T. D. Noe, Feb 12 2004


STATUS

approved



