A055781 Primes q of the form q = 10p + 1, where p is also prime.

31, 71, 131, 191, 311, 431, 971, 1031, 1091, 1511, 1571, 1811, 1931, 2111, 2411, 2711, 3371, 3491, 3671, 4091, 4211, 4391, 4871, 5231, 5471, 5711, 6011, 6131, 6311, 6911, 7331, 7691, 8111, 8231, 8291, 8831, 9371, 10091, 10211, 10331, 10391, 10631

Offset: 1

Author

Labos Elemer, Jul 13 2000

Keywords

• nonn

Comments

Corresponding values of p in A023237. - Jaroslav Krizek, Jul 14 2010

From Sergey Pavlov, Jun 14 2017: (Start)

Let a, b, and c be prime numbers such that c = 10b + 1 = 10 * (10a + 1) + 1. Then c = 311, b = 31, a = 3. (There are no other solutions since any prime p > 3 is either of the form 3k + 1 or 3k - 1. In other words, while a > 3 and a, b are primes, a == 1 (mod 3), b == -1 (mod 3), whereas c == 0 (mod 3).)

So is for any similar sequence of primes (of the form kn + 1) where 2k + 1 == 0 (mod 3), e.g., for A002144: the equation of the form c = kb + 1 = k * (ka + 1) + 1 while a, b, c are primes could have the only solution iff a = 3 (but also could have not).

(End) [This comment needs to be rewritten. - N. J. A. Sloane, Feb 18 2019]

Examples

			1031 = 103*10 + 1, 1 appended to 103.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005384, A005385, A023237.

Programs

Maple

select(isprime, map(t-> 10*t+1, select(isprime, [3,seq(i,i=7..2000,6)]))); # Robert Israel, Jun 13 2017

Mathematica

Select[10Prime[Range[200]]+1,PrimeQ]  (* Harvey P. Dale, Feb 04 2011 *)

PARI

is(n)=n%10==1 && isprime(n) && isprime(n\10) \\ Charles R Greathouse IV, Jun 17 2017