A006562 Balanced primes (of order one): primes which are the average of the previous prime and the following prime.
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
Offset: 1
Examples
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).
References
- A. Murthy, Smarandache Notions Journal, Vol. 11 N. 1-2-3 Spring 2000.
- 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.
- Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
- 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 triple, 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.
- Wikipedia, Balanced prime.
Crossrefs
Primes A000040 whose indices are 1 more than A064113, indices of zeros in A036263 (second differences of the primes).
Programs
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Haskell
a006562 n = a006562_list !! (n-1) a006562_list = filter ((== 1) . a010051) a075540_list -- Reinhard Zumkeller, Jan 20 2012
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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
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Magma
[a: n in [1..1000] | IsPrime(a) where a is NthPrime(n)-NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016
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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]]]] Prime[#]&/@SequencePosition[Differences[Prime[Range[800]]],{x_,x_}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 31 2019 *) -
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 -
PARI
forprime(p=1,999, p-precprime(n-1)==nextprime(p+1)-p && print1(p",")) \\ M. F. Hasler, Jun 01 2013
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PARI
is(n)=n-precprime(n-1)==nextprime(n+1)-n && isprime(n) \\ Charles R Greathouse IV, Apr 07 2016
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Python
from sympy import nextprime; p, q, r = 2, 3, 5 while q < 6000: if 2*q == p + r: print(q, end = ", ") p, q, r = q, r, nextprime(r) # Ya-Ping Lu, Dec 23 2021
Formula
2*p_n = p_(n-1) + p_(n+1).
Equals { p = prime(k) | A118534(k) = prime(k-1) }. - Rémi Eismann, Nov 30 2009
Conjecture: Limit_{n->oo} n*(log(a(n)))^2 / a(n) = 1/2. - Alain Rocchelli, Mar 21 2024
Conjecture: The asymptotic limit of the average of a(n+1)-a(n) is equivalent to 2*(log(a(n)))^2. Otherwise formulated: 2 * Sum_{n=1..N} (log(a(n)))^2 ~ a(N). - Alain Rocchelli, Mar 23 2024
Extensions
Reworded comment and added formula from R. Eismann. - M. F. Hasler, Nov 30 2009
Edited by Daniel Forgues, Jan 15 2011
Comments