%I A006562 M4011
%S A006562 5,53,157,173,211,257,263,373,563,593,607,653,733,947,977,1103,1123,
%T A006562 1187,1223,1367,1511,1747,1753,1907,2287,2417,2677,2903,2963,3307,3313,
%U A006562 3637,3733,4013,4409,4457,4597,4657,4691,4993,5107,5113,5303,5387,5393
%N A006562 Balanced primes (of order one): primes which are the average of the previous 
               prime and the following prime.
%C A006562 Subsequence of A075540. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 11 2006
%C A006562 This subsequence of A125830 and of A162174 gives primes of level (1,1): 
               More generally, the i-th prime p(i) is of level (1,k) iff it has 
               level 1 in A117563 and 2 p(i) - p(i+1) = p(i-k). - Remi Eismann (reismann(AT)free.fr), 
               Feb 15 2007
%C A006562 Note the similarity between plots of A006562 and A013916 [From Bill McEachen 
               (bmceache(AT)centralsan.org), Sep 07 2009]
%D A006562 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006562 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 870.
%D A006562 A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.
%D A006562 David Wells, The Penguin Dictionary of Curious and Interesting Numbers 
               (Rev. ed. 1997), p. 134
%H A006562 T. D. Noe, <a href="b006562.txt">Table of n, a(n) for n=1..10000</a>
%H A006562 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%F A006562 2*p_n = p_(n-1) + p_(n+1).
%F A006562 A006562 = { p = prime(k) | A118534(k) = prime(k-1) }. [From R. Eismann, 
               Nov 30 2009]
%e A006562 5 belongs to the sequence because 5 = (3 + 7)/2. Likewise 53 = (47 + 
               59)/2.
%t A006562 Transpose[ Select[ Partition[ Prime[ Range[1000]], 3, 1], #[[2]] ==(#[[1]] 
               + #[[3]])/2 &]][[2]]
%o A006562 (PARI) betwixtpr(n) = { local(c1,c2,x,y); for(x=2,n, c1=c2=0; for(y=prime(x-1)+1,
               prime(x)-1, if(!isprime(y),c1++); ); for(y=prime(x)+1,prime(x+1)-1, 
               if(!isprime(y),c2++); ); if(c1==c2,print1(prime(x)",")) ) } - Cino 
               Hilliard (hillcino368(AT)gmail.com), Jan 25 2005
%Y A006562 Cf. A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, 
               A096702, A096703, A096704.
%Y A006562 Cf. A096693, A051634, A051635, A054342.
%Y A006562 Cf. A117078, A117563, A125830, A006562, A117876, A125576
%Y A006562 Sequence in context: A106097 A163580 A075540 this_sequence A094847 A001992 
               A139899
%Y A006562 Adjacent sequences: A006559 A006560 A006561 this_sequence A006563 A006564 
               A006565
%K A006562 nonn,easy,nice,new
%O A006562 1,1
%A A006562 N. J. A. Sloane (njas(AT)research.att.com) and Robert G. Wilson v
%E A006562 Reworded comment and added formula from R. Eismann M. F. Hasler (mhasler(AT)univ-ag.fr), 
               Nov 30 2009