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 A006562 Balanced primes (of order one): primes which are the average of the previous prime and the following prime. (Formerly M4011) 128

%I M4011

%S 5,53,157,173,211,257,263,373,563,593,607,653,733,947,977,1103,1123,

%T 1187,1223,1367,1511,1747,1753,1907,2287,2417,2677,2903,2963,3307,

%U 3313,3637,3733,4013,4409,4457,4597,4657,4691,4993,5107,5113,5303,5387,5393

%N Balanced primes (of order one): primes which are the average of the previous prime and the following prime.

%C Subsequence of A075540. - _Franklin T. Adams-Watters_, Jan 11 2006

%C 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) if and only if it has level 1 in A117563 and 2 p(i) - p(i+1) = p(i-k). - _Rémi Eismann_, Feb 15 2007

%C Note the similarity between plots of A006562 and A013916. - _Bill McEachen_, Sep 07 2009

%C Balanced primes U strong primes = good primes. Or, A006562 U A051634 = A046869. - _Juri-Stepan Gerasimov_, Mar 01 2010

%C Primes prime(n) such that A001223(n-1) = A001223(n). - _Irina Gerasimova_, Jul 11 2013

%D A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134.

%H T. D. Noe, <a href="/A006562/b006562.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.

%H Shubhankar Paul, <a href="https://www.erpublication.org/admin/vol_issue1/upload%20Image/IJETR011954.pdf">Ten Problems of Number Theory</a>, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.

%H Shubhankar Paul, <a href="http://erpublication.org/admin/vol_issue1/upload%20Image/IJETR012013.pdf">Legendre, Grimm, Balanced Prime, Prime triple, Polignac's conjecture, a problem and 17 tips with proof to solve problems on number theory</a>, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-10, December 2013.

%F 2*p_n = p_(n-1) + p_(n+1).

%F Equals { p = prime(k) | A118534(k) = prime(k-1) }. - _Rémi Eismann_, Nov 30 2009

%F a(n) = A000040(A064113(n) + 1) = (A122535(n) + A181424(n)) / 2. - _Reinhard Zumkeller_, Jan 20 2012

%F a(n) = A122535(n) + A117217(n). - _Zak Seidov_, Feb 14 2013

%F Equals A145025 intersect A000040 = A145025 \ A024675. - _M. F. Hasler_, Jun 01 2013

%e 5 belongs to the sequence because 5 = (3 + 7)/2. Likewise 53 = (47 + 59)/2.

%e 5 belongs to the sequence because it is a term, but not first or last, of the AP of consecutive primes (3, 5, 7).

%e 53 belongs to the sequence because it is a term, but not first or last, of the AP of consecutive primes (47, 53, 59).

%e 257 and 263 belong to the sequence because they are terms, but not first or last, of the AP of consecutive primes (251, 257, 263, 269).

%p A006562:=proc(q)

%p local a,b,c,n;

%p for n from 2 to q do

%p a:=ithprime(n); b:=prevprime(a); c:=nextprime(a);

%p if a-b=c-a then print(a); fi; od; end:

%p A006562(10000); # _Paolo P. Lava_, Nov 08 2012

%t Transpose[ Select[ Partition[ Prime[ Range], 3, 1], #[] ==(#[] + #[])/2 &]][]

%t p=Prime[Range]; p[[Flatten[1+Position[Differences[p, 2], 0]]]]

%t Prime[#]&/@SequencePosition[Differences[Prime[Range]],{x_,x_}][[All,2]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 31 2019 *)

%o (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_, Jan 25 2005

%o (PARI) forprime(n=1,999,n-precprime(n-1)==nextprime(n+1)-n&print1(n",")) \\ _M. F. Hasler_, Jun 01 2013

%o (PARI) is(n)=n-precprime(n-1)==nextprime(n+1)-n && isprime(n) \\ _Charles R Greathouse IV_, Apr 07 2016

%o a006562 n = a006562_list !! (n-1)

%o a006562_list = filter ((== 1) . a010051) a075540_list

%o -- _Reinhard Zumkeller_, Jan 20 2012

%o a006562 n = a006562_list !! (n-1)

%o a006562_list = h a000040_list where

%o h (p:qs@(q:r:ps)) = if 2 * q == (p + r) then q : h qs else h qs

%o -- _Reinhard Zumkeller_, May 09 2013

%o (MAGMA) [a: n in [1..1000] | IsPrime(a) where a is NthPrime(n)-NthPrime(n+1)+NthPrime(n+2)]; // _Vincenzo Librandi_, Jun 23 2016

%Y Cf. A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096693, A051634, A051635, A054342, A117078, A117563, A125830, A117876, A125576, A046869, A173891, A173892, A173893, A006560, A075540.

%Y Cf. A225494 (multiplicative closure); complement of A178943 with respect to A000040.

%Y Cf. A055380, A051795, A081415, A096710 for other balanced prime sequences.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_ and _Robert G. Wilson v_

%E Reworded comment and added formula from R. Eismann. - _M. F. Hasler_, Nov 30 2009

%E Edited by _Daniel Forgues_, Jan 15 2011

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)