A178943 -id:A178943 - cp's OEIS frontend

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

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

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
Primes which are the average of three consecutive primes. - Peter Schorn, Apr 30 2025

			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).

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.

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.

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.

a006562 n = a006562_list !! (n-1)
a006562_list = filter ((== 1) . a010051) a075540_list
-- Reinhard Zumkeller, Jan 20 2012

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

[a: n in [1..1000] | IsPrime(a) where a is NthPrime(n)-NthPrime(n+1)+NthPrime(n+2)]; // Vincenzo Librandi, Jun 23 2016

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 *)

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

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

is(n)=n-precprime(n-1)==nextprime(n+1)-n && isprime(n) \\ Charles R Greathouse IV, Apr 07 2016

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

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

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

Edited by Daniel Forgues, Jan 15 2011

A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.

Original entry on oeis.org

3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523, 547, 571, 577, 601, 619, 643, 647

Offset: 1

Felice Russo, Nov 15 1999

Primes prime(n) such that prime(n)-prime(n-1) < prime(n+1)-prime(n). - Juri-Stepan Gerasimov, Jan 01 2011
a(n) < A051634(n). a(n) ~ 2*prime(n). - Thomas Ordowski, Jul 25 2012
The inequality above is false. The least counterexample is a(19799) = 496291 > A051634(19799) = 496283. - Amiram Eldar, Nov 26 2023
Erdős called a weak prime an "early prime." He conjectured that there are infinitely many consecutive pairs of early primes, and offered $100 for a proof and $25000 for a disproof (Kuperberg 1992). See A229832 for a stronger conjecture. - Jonathan Sondow, Oct 13 2013

			7 belongs to the sequence because 7 < (5+11)/2.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Greg Kuperberg, The Erdos kitty: At least $9050 in prizes!, Newsgroups: rec.puzzles, sci.math, 1992. [Broken link]
Greg Kuperberg, The Erdos kitty: At least $9050 in prizes!, Newsgroups: rec.puzzles, sci.math, 1992. [Cached copy]
Wikipedia, Weak prime.

Cf. A006562, A051634, A229832.
Subsequence of A178943.
Cf. A225495 (multiplicative closure).

a051635 n = a051635_list !! (n-1)
a051635_list = g a000040_list where
   g (p:qs@(q:r:ps)) = if 2 * q < (p + r) then q : g qs else g qs
-- Reinhard Zumkeller, May 09 2013

Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]<(#[[1]]+#[[3]])/2 &]][[2]] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p, 2]], 1]]]]

p=2;q=3;forprime(r=5,1e3,if(2*qCharles R Greathouse IV, Jul 25 2011

a(1) = A229832(1). - Jonathan Sondow, Oct 13 2013

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024

More terms from James Sellers

A051634 Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.

Original entry on oeis.org

11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 569, 587, 599, 613, 617, 631, 641, 659, 673, 701

Offset: 1

Felice Russo, Nov 15 1999

Prime(k) such that prime(k) - prime(k-1) > prime(k+1) - prime(k). - Juri-Stepan Gerasimov, Jan 01 2011
a(n) > A051635(n). - Thomas Ordowski, Jul 25 2012
The inequality above is false. The least counterexample is a(19799) = 496283 < A051635(19799) = 496291. - Amiram Eldar, Nov 26 2023
Conjecture: Limit_{N->oo} Sum_{n=1..N} (NextPrime(a(n))-a(n)) / a(N) = 1/4. [A heuristic proof is available at www.primepuzzles.net - Conjecture 91] - Alain Rocchelli, Nov 14 2022
A131499 is a subsequence. - Davide Rotondo, Oct 16 2023

			11 belongs to the sequence because 11 > (7 + 13)/2.

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Carlos Rivera, Conjecture 91. A conjecture about strong primes, The Prime Puzzles & Problems Connection.

Cf. A006562, A051635, A229832.
Subsequence of A178943.
Cf. A225493 (multiplicative closure), A131499 (subsequence).

a051634 n = a051634_list !! (n-1)
a051634_list = f a000040_list where
   f (p:qs@(q:r:ps)) = if 2 * q > (p + r) then q : f qs else f qs
-- Reinhard Zumkeller, May 09 2013

q:= n-> isprime(n) and 2*n>prevprime(n)+nextprime(n):
select(q, [$3..1000])[];  # Alois P. Heinz, Jun 21 2023

Transpose[Select[Partition[Prime[Range[10^2]], 3, 1], #[[2]]>(#[[1]]+#[[3]])/2 &]][[2]] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
p=Prime[Range[200]]; p[[Flatten[1+Position[Sign[Differences[p,2]], -1]]]]

p=2;q=3;forprime(r=5,1e4,if(2*q>p+r,print1(q", "));p=q;q=r) \\ Charles R Greathouse IV, Jul 19 2011

from sympy import nextprime
def aupto(limit):
    alst, p, q, r = [], 2, 3, 5
    while q <= limit:
        if 2*q > p + r: alst.append(q)
        p, q, r = q, r, nextprime(r)
    return alst
print(aupto(701)) # Michael S. Branicky, Nov 17 2021

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = 1/2. - Alain Rocchelli, Mar 17 2024

A225496 Numbers having no balanced prime factors, cf. A006562.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Reinhard Zumkeller, May 09 2013

nonn

a(n) = A047201(n) for n <= 42.

			a(40) = 49 = 7^2 = A178943(3)^2;
a(41) = 51 = 3 * 17 = A178943(2) * A178943(6);
a(42) = 52 = 2^2 * 13 = A178943(1)^2 * A178943(5);
a(43) = 54 = 2 * 3^3 = A178943(1) * A178943(2)^3;
a(44) = 56 = 2^3 * 7 = A178943(1)^3 * A178943(3);
a(45) = 57 = 3 * 19 = A178943(2) * A178943(7).

Reinhard Zumkeller, Table of n, a(n) for n = 1..5000

Cf. A225493 (strong), A225494 (balanced), A225495 (weak).

import Data.Set (singleton, fromList, union, deleteFindMin)
a225496 n = a225496_list !! (n-1)
a225496_list = 1 : h (singleton p) ps [p] where
   (p:ps) = a178943_list
   h s xs'@(x:xs) ys
     | m > x     = h (s `union` (fromList $ map (* x) (1 : ys))) xs ys
     | otherwise = m : h (s' `union` (fromList $ map (* m) ys')) xs' ys'
     where ys' = m : ys; (m, s') = deleteFindMin s

Multiplicative closure of A178943; a(n) mod A006562(k) > 0 for all k.

cp's OEIS Frontend

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

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A051635 Weak primes: prime(n) < (prime(n-1) + prime(n+1))/2.

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A051634 Strong primes: prime(k) > (prime(k-1) + prime(k+1))/2.

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A225496 Numbers having no balanced prime factors, cf. A006562.

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