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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)
111
5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subsequence of A075540. - Franklin T. Adams-Watters, Jan 11 2006

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

Note the similarity between plots of A006562 and A013916. [Bill McEachen, Sep 07 2009]

Balanced primes U strong primes = good primes. Or, A006562 U A051634 = A046869. [From Juri-Stepan Gerasimov, Mar 01 2010]

A prime in an arithmetic progression of 3 or more consecutive primes, excepting the first and last primes of said progression. - Daniel Forgues, Jan 11 2011

Primes prime(n) such that A001223(n-1) = A001223(n). - Irina Gerasimova, Jul 11 2013

REFERENCES

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

Shubhankar Paul, Ten Problems of Number Theory, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013

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

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

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 870.

FORMULA

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

A006562 = { p = prime(k) | A118534(k) = prime(k-1) }. [R. Eismann, Nov 30 2009]

a(n) = A000040(A064113(n) + 1) = (A122535(n) + A181424(n)) / 2. [Reinhard Zumkeller, Jan 20 2012]

a(n) = A122535(n) + A117217(n). - Zak Seidov, Feb 14 2013

A006562 = A145025 intersect A000040 = A145025 \ A024675. - M. F. Hasler, Jun 01 2013

EXAMPLE

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

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

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

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).

MAPLE

A006562:=proc(q)

local a, b, c, n;

for n from 2 to q do

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

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

A006562(10000); # Paolo P. Lava, Nov 08 2012

MATHEMATICA

Transpose[ Select[ Partition[ Prime[ Range[1000]], 3, 1], #[[2]] ==(#[[1]] + #[[3]])/2 &]][[2]]

p=Prime[Range[1000]]; p[[Flatten[1+Position[Differences[p, 2], 0]]]]

PROG

(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 */

(Haskell)

a006562 n = a006562_list !! (n-1)

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

-- Reinhard Zumkeller, Jan 20 2012

(Haskell)

a006562 n = a006562_list !! (n-1)

a006562_list = h a000040_list where

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

-- Reinhard Zumkeller, May 09 2013

(PARI) forprime(n=1, 999, n-precprime(n-1)==nextprime(n+1)-n&print1(n", ")) \\ - M. F. Hasler, Jun 01 2013

CROSSREFS

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.

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

Sequence in context: A106097 A163580 A075540 * A094847 A001992 A139899

Adjacent sequences:  A006559 A006560 A006561 * A006563 A006564 A006565

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane and Robert G. Wilson v

EXTENSIONS

Reworded comment and added formula from R. Eismann. - M. F. Hasler, Nov 30 2009

Edited by Daniel Forgues, Jan 15 2011

STATUS

approved

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Last modified October 25 22:07 EDT 2014. Contains 248558 sequences.