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

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

1,1

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. - Juri-Stepan Gerasimov, Mar 01 2010

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

Numbers m such that A346399(m) is odd and >= 3. - Ya-Ping Lu, Dec 26 2021 and May 07 2024

"Balanced" means that the next and preceding gap are of the same size, i.e., the second difference A036263 vanishes; so these are the primes whose indices are 1 more than indices of zeros in A036263, listed in A064113. - M. F. Hasler, Oct 15 2024

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.

Formula

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

Equals { p = prime(k) | A118534(k) = prime(k-1) }. - Rémi 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

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

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

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).

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 *)

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
(PARI) forprime(p=1, 999, p-precprime(n-1)==nextprime(p+1)-p && print1(p", ")) \\ M. F. Hasler, Jun 01 2013
(PARI) is(n)=n-precprime(n-1)==nextprime(n+1)-n && isprime(n) \\ Charles R Greathouse IV, Apr 07 2016
(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
(Magma) [a: n in [1..1000] | IsPrime(a) where a is NthPrime(n)-NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016
(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

Crossrefs

Primes A000040 whose indices are 1 more than A064113, indices of zeros in A036263 (second differences of the primes).

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.

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

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