A029707 Numbers n such that the n-th and the (n+1)-st primes are twin primes.

2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, 57, 60, 64, 69, 81, 83, 89, 98, 104, 109, 113, 116, 120, 140, 142, 144, 148, 152, 171, 173, 176, 178, 182, 190, 201, 206, 209, 212, 215, 225, 230, 234, 236, 253, 256, 262, 265, 268, 277

Offset: 1

Author

N. J. A. Sloane, Dec 11 1999

Keywords

• nonn

Comments

Numbers m such that prime(m)^2 == 1 mod (prime(m) + prime(m + 1)). - Zak Seidov, Sep 18 2013

First differences are A027833. The complement is A049579. - Gus Wiseman, Dec 03 2024

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..86027
• Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Crossrefs

Cf. A014574, A027833 (first differences), A007508. Equals PrimePi(A001359) (cf. A000720).

The complement is A049579, first differences A251092 except first term.

Lengths of runs of terms differing by 2 are A179067.

The first differences have run-lengths A373820 except first term.

A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).

A038664 finds the first prime gap of 2n.

A046933 counts composite numbers between primes.

For prime runs: A005381, A006512, A025584, A067774.

Cf. A006560, A006562, A037201, A107770, A122535, A155752, A175632, A176246, A373819.

Programs

Maple

A029707 := proc(n)
    numtheory[pi](A001359(n)) ;
end proc:
seq(A029707(n),n=1..30); # R. J. Mathar, Feb 19 2017

Mathematica

Select[ Range@300, PrimeQ[ Prime@# + 2] &] (* Robert G. Wilson v, Mar 11 2007 *)
Flatten[Position[Flatten[Differences/@Partition[Prime[Range[100]],2,1]], 2]](* Harvey P. Dale, Jun 05 2014 *)

Sage

def A029707(n) :
   a = [ ]
   for i in (1..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i)
   return(a)
A029707(277) # Jani Melik, May 15 2014

Formula

a(n) = A107770(n) - 1. - Juri-Stepan Gerasimov, Dec 16 2009