cp's OEIS Frontend

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

%I A029707 #56 Dec 04 2024 21:24:04
%S A029707 2,3,5,7,10,13,17,20,26,28,33,35,41,43,45,49,52,57,60,64,69,81,83,89,
%T A029707 98,104,109,113,116,120,140,142,144,148,152,171,173,176,178,182,190,
%U A029707 201,206,209,212,215,225,230,234,236,253,256,262,265,268,277
%N A029707 Numbers n such that the n-th and the (n+1)-st primes are twin primes.
%C A029707 Numbers m such that prime(m)^2 == 1 mod (prime(m) + prime(m + 1)). - _Zak Seidov_, Sep 18 2013
%C A029707 First differences are A027833. The complement is A049579. - _Gus Wiseman_, Dec 03 2024
%H A029707 Robert G. Wilson v, <a href="/A029707/b029707.txt">Table of n, a(n) for n = 1..86027</a>
%H A029707 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%F A029707 a(n) = A107770(n) - 1. - _Juri-Stepan Gerasimov_, Dec 16 2009
%p A029707 A029707 := proc(n)
%p A029707     numtheory[pi](A001359(n)) ;
%p A029707 end proc:
%p A029707 seq(A029707(n),n=1..30); # _R. J. Mathar_, Feb 19 2017
%t A029707 Select[ Range@300, PrimeQ[ Prime@# + 2] &] (* _Robert G. Wilson v_, Mar 11 2007 *)
%t A029707 Flatten[Position[Flatten[Differences/@Partition[Prime[Range[100]],2,1]], 2]](* _Harvey P. Dale_, Jun 05 2014 *)
%o A029707 (Sage)
%o A029707 def A029707(n) :
%o A029707    a = [ ]
%o A029707    for i in (1..n) :
%o A029707       if (nth_prime(i+1)-nth_prime(i) == 2) :
%o A029707          a.append(i)
%o A029707    return(a)
%o A029707 A029707(277) # _Jani Melik_, May 15 2014
%Y A029707 Cf. A014574, A027833 (first differences), A007508. Equals PrimePi(A001359) (cf. A000720).
%Y A029707 The complement is A049579, first differences A251092 except first term.
%Y A029707 Lengths of runs of terms differing by 2 are A179067.
%Y A029707 The first differences have run-lengths A373820 except first term.
%Y A029707 A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
%Y A029707 A038664 finds the first prime gap of 2n.
%Y A029707 A046933 counts composite numbers between primes.
%Y A029707 For prime runs: A005381, A006512, A025584, A067774.
%Y A029707 Cf. A006560, A006562, A037201, A107770, A122535, A155752, A175632, A176246, A373819.
%K A029707 nonn
%O A029707 1,1
%A A029707 _N. J. A. Sloane_, Dec 11 1999