A096710 -id:A096710 - 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

A055380 Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.

Original entry on oeis.org

5, 18731, 683783, 98303927, 60335249959, 1169769749219, 3945769040699039, 159067808851610657, 6919940122097246597

Offset: 1

Jud McCranie, Jun 23 2000

Least n-tuply balanced primes: primes which are averages of both their immediate neighbors, their second neighbors, their third neighbors, ... and their n-th neighbors.
a(9) <= 6919940122097246597. The solution was found by the BOINC project "SPT test project". - Natalia Makarova, Nov 25 2023
a(n) is the smallest number m such that A346399(m) = 2n + 1. - Ya-Ping Lu, May 12 2024

			In 5-tuple of consecutive primes (18713, 18719, 18731, 18743, 18749), the primes are symmetric w.r.t. its central prime 18731, since 18713+18749 = 18719+18743 = 2*18731, and this is the smallest such 5-tuple. Hence, a(2)=18731.
Alternatively, the symmetry can be seen from the differences between consecutive primes. For (18713, 18719, 18731, 18743, 18749), the differences are (6,12,12,6).

Stop@home, BOINC project to search all up to 2^64. [Dead link]
Symmetric Prime Tuples, SPT test project.

Cf. A001223, A055381, A055382, A006562, A051795, A081415, A096710, A346399.

Table[i = n + 2;
 While[x = Differences[Table[Prime[k + i], {k, -n, n}]];
x != Reverse[x], i++]; Prime[i], {n, 3}] (* Robert Price, Oct 12 2019 *)

a(n) = A151800^(n)(A175309(2n)), i.e., A151800 applied n times on A175309(2n). - Max Alekseyev, Jul 26 2014

a(6) from Donovan Johnson, Mar 09 2008
Definition corrected by Max Alekseyev, Jul 29 2014
a(7) from Dmitry Petukhov, added by Max Alekseyev, Nov 03 2014
a(8) from SPT project, added by Dmitry Petukhov, Apr 06 2017
a(9) from SPT project, added by Dmitry Petukhov, Mar 25 2025

A082079 Balanced primes of order four.

Original entry on oeis.org

491, 757, 1787, 3571, 6337, 6451, 6991, 7741, 7907, 8821, 10141, 10267, 10657, 12911, 15299, 16189, 18223, 18701, 19801, 19843, 19853, 19937, 21961, 22543, 22739, 22807, 23893, 23909, 24767, 25169, 25391, 26591, 26641, 26693, 26713

Offset: 1

Labos Elemer, Apr 08 2003

nonn

The arithmetic mean of 8 primes in its "neighborhood"; not to be confused with 'Quadruply balanced primes' (A096710).

A balanced prime of order four is not necessarily balanced (A006562) order one, or of order two (A082077), or of order three (A082078), etc.

			p = 491 = (463 + 467 + 479 + 487 + 491 + 499 + 503 + 509 + 521)/9 = 4419/9.

Aaron Toponce, Table of n, a(n) for n = 1..1000

Cf. A006562, A082077, A082078, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704.

Cf. A096693, A082080, A081415, A051795, A006562.

P:=Filtered([1..50000],IsPrime);;
a:=List(Filtered(List([0..3000],k->List([5..13],j->P[j-4+k])), i-> Sum(i)/9=i[5]),m->m[5]); # Muniru A Asiru, Feb 14 2018

Do[s3=Prime[n]+Prime[n+1]+Prime[n+2]; s5=Prime[n-1]+s3+Prime[n+3]; s7=Prime[n-2]+s5+Prime[n+4]; s9=Prime[n-3]+s7+Prime[n+5]; If[Equal[s9/9, Prime[n+1]], Print[Prime[n+1]]], {n, 4, 10000}]
(* Second program: *)
With[{k = 4}, Select[MapIndexed[{Prime[First@ #2 + k], #1} &, Mean /@ Partition[Prime@ Range[3000], 2 k + 1, 1]], SameQ @@ # &][[All, 1]]] (* Michael De Vlieger, Feb 15 2018 *)
Select[Partition[Prime[Range[3000]],9,1],Mean[#]==#[[5]]&][[;;,5]] (* Harvey P. Dale, Mar 09 2023 *)

isok(p) = {if (isprime(p), k = primepi(p); if (k > 4, sum(i=k-4, k+4, prime(i)) == 9*p;););} \\ Michel Marcus, Mar 07 2018

A051795 Doubly balanced primes: primes which are averages of both their immediate and their second neighbors.

Original entry on oeis.org

18731, 25621, 28069, 30059, 31051, 44741, 76913, 97441, 103669, 106681, 118831, 128449, 135089, 182549, 202999, 240491, 245771, 249199, 267569, 295387, 347329, 372751, 381401, 435751, 451337, 455419, 471521, 478099, 498301, 516877, 526441, 575231, 577873

Offset: 1

Harvey P. Dale, Dec 10 1999

Could also be called overbalanced or [3,5]-balanced primes: balanced primes which are equally average of 3,5 consecutive prime neighbors as follows: a(n)=[q+a(n)+r]/3=[p+q+a(n)+r+s]/5 See 3-balanced=A006562;[3,5,7]-balanced=A081415. - Labos Elemer, Apr 02 2003

Numbers m such that A346399(m) is odd and >= 5. - Ya-Ping Lu, May 11 2024

			25621 belongs to the sequence because 25621 = (25609 + 25633)/2 = (25603 + 25609 + 25633 + 25639)/4.

Jud McCranie and Sebastian Petzelberger, Table of n, a(n) for n = 1..10000 (first 1000 terms from Jud McCranie)

Cf. A006562, A081415, A096710, A055380, A346399.

Transpose[Select[Partition[Prime[Range[50000]],5,1],(#[[1]]+#[[5]])/2 == (#[[2]]+#[[4]])/2 == #[[3]]&]][[3]] (* Harvey P. Dale, Sep 13 2013 *)

from sympy import nextprime; p, q, r, s, t = 2, 3, 5, 7, 11
while t < 580000:
    if p + t == q + s == 2*r: print(r, end = ', ')
    p, q, r, s, t = q, r, s, t, nextprime(t) # Ya-Ping Lu, May 11 2024

A175309 a(n) = the smallest prime prime(k) such that prime(k+j) - prime(k+j-1) = prime(n+k+1-j) - prime(n+k-j) for all j with 1 <= j <= n.

Original entry on oeis.org

2, 3, 5, 18713, 5, 683747, 17, 98303867, 13, 60335249851, 137, 1169769749111, 8021749, 3945769040698829, 1071065111, 159067808851610411, 1613902553, 6919940122097246303, 1797595814863

Offset: 1

Leroy Quet, Mar 27 2010

From M. F. Hasler, Apr 02 2010: (Start)
Also: Start of the first sequence of n+1 consecutive primes symmetrically distributed w.r.t. their barycenter, e.g., [2,3], [3,5,7], [5,7,11,13], [18713, 18719, 18731, 18743, 18749]. With this definition, it would make sense to prefix the sequence with an initial term a(0)=2.
Sequence A081235 (or A055382, which is essentially the same) consists of every other term of this sequence. (End)
a(19) = 1797595814863, a(21) = 633925574060671, a(23) = 22930603692243271. - Tomáš Brada, May 25 2020

Andrew Howroyd, Table of n, a(n) for n = 1..19
BOINC project to search all up to 2^64
Symmetric Prime Tuples, SPT test project

Cf. A006562, A051795, A081235, A081415, A096710, A055382.

A175309[n_] := Module[{k},
   k = 1; While[! AllTrue[Range[n], Prime[k+#] - Prime[k+#-1] ==
        Prime[n+k+1-#] - Prime[n+k-#] &], k++]; Return[Prime[k]]];
Table[A175309[n], {n, 1, 7}]  (* Robert Price, Mar 27 2019 *)

a(n)={ my( last=vector(n++,i,prime(i)), m, i=Mod(n-2,n)); forprime(p=last[n],default(primelimit), m=last[1+lift(i+2)]+last[1+lift(i++)]=p; for( j=1,n\2, last[1+lift(i-j)]+last[1+lift(i+j+1)]==m || next(2)); return( last[1+lift(i+1)])) } \\ M. F. Hasler, Apr 02 2010

isok(p, n) = {my(k=primepi(p)); for (j=1, n, if (prime(k+j) - prime(k+j-1) != prime(n+k+1-j) - prime(n+k-j), return (0));); return (1);} \\ Michel Marcus, Apr 08 2017

a(2n-1) = A081235(n) (= A055382(n) for n>1). - M. F. Hasler, Apr 02 2010

Terms through a(12) were calculated by (in alphabetical order) Franklin T. Adams-Watters, Hans Havermann and D. S. McNeil
Minor edits by N. J. A. Sloane, Apr 02 2010
a(14) from Dmitry Petukhov, added by Max Alekseyev, Nov 03 2014
a(16) from BOINC project, added by Dmitry Petukhov, Apr 06 2017
a(18)-a(19) from SPT test project, added by Dmitry Petukhov, Mar 16 2025

A081415 Triply balanced primes: primes which are averages of both their immediate neighbor, their second neighbors and their third neighbors.

Original entry on oeis.org

683783, 1056317, 1100261, 2241709, 2815301, 4746359, 10009049, 12003209, 13810981, 14907649, 15403009, 15730067, 16595081, 17518201, 19755301, 20378327, 21006487, 21574453, 21579983, 22237121, 22625179, 25876901, 26018791, 26354201, 27188141, 28469461

Offset: 1

Labos Elemer, Apr 02 2003

nonn

Equivalently, primes which are balanced primes of orders 1, 2, and 3. - Muniru A Asiru, Apr 08 2018

Numbers m such that A346399(m) is odd and >= 7. - Ya-Ping Lu, May 11 2024

			p = 683383: 683747 + ... + p + ... + 683819 = 7p; 683759 + ... + p + ... + 683807 = 5p; 683777 + p + 683789 = 3p.

Jud McCranie, Table of n, a(n) for n = 1..1000
Wikipedia, Balanced prime

Cf. A006562, A051795, A055380, A096710, A346399.

P:=Filtered([1,3..3*10^7+1],IsPrime);;
a:=Intersection(List([1,2,3],b->List(Filtered(List([0..Length(P)-(2*b+1)],k->List([1..2*b+1],j->P[j+k])),i->Sum(i)/(2*b+1)=i[b+1]),m->m[b+1]))); # Muniru A Asiru, Apr 08 2018

a = {}; Do[p = 2Prime[n]; If[p == Prime[n - 1] + Prime[n + 1] && p == Prime[n - 2] + Prime[n + 2] && p == Prime[n - 3] + Prime[n + 3], Print[p / 2]; AppendTo[a, p / 2]], {n, 5, 1100000}]; a (* Robert G. Wilson v, Jun 28 2004 *)
Transpose[Select[Partition[Prime[Range[1620000]],7,1],(#[[1]]+#[[7]])/2 == (#[[2]]+#[[6]])/2==(#[[3]]+#[[5]])/2==#[[4]]&]][[4]] (* Harvey P. Dale, Sep 13 2013 *)

from sympy import nextprime; p, q, r, s, t, u, v = 2, 3, 5, 7, 11, 13, 17
while v < 29000000:
    if p + v == q + u == r + t == 2*s: print(s, end = ', ')
    p, q, r, s, t, u, v = q, r, s, t, u, v, nextprime(v) # Ya-Ping Lu, May 11 2024

A082080 Smallest balanced prime of order n.

Original entry on oeis.org

2, 5, 79, 17, 491, 53, 71, 29, 37, 983, 5503, 173, 157, 353, 5297, 263, 179, 383, 137, 2939, 2083, 751, 353, 5501, 1523, 149, 4561, 1259, 397, 787, 8803, 8803, 607, 227, 3671, 17443, 57097, 3607, 23671, 12539, 1217, 11087, 1087, 21407, 19759, 953

Offset: 0

Labos Elemer, Apr 08 2003

nonn

Or, smallest (2n+1)-balanced prime number.

Prime(k) is a balanced prime of order n if it is the average of the 2n+1 primes from prime(k-n) to prime(k+n).

			a(1) = 5 = (3 + 5 + 7)/3 = 15/3.
a(5) = 53 = (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73)/11 = 583/11.
a(6) = 71 = (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101)/13 = 923/13.

Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 1375 terms from Robert G. Wilson v)

Cf. A096693, A006562, A082077, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704.

Cf. A006562, A051795, A081415, A096710, A055380, A082312, A075540, A054800.

f[n_] := Block[{p = Prime@ Range[2n +1]}, While[ Total[p] != (2n +1) p[[n +1]], p = Join[Rest@ p, {NextPrime[ p[[-1]]] }]]; p[[n +1]]]; Array[f, 46, 0] (* Robert G. Wilson v, Jun 21 2004 and modified Apr 11 2017 *)

for(n=0, 50, i=2*n+1;f=0;forprime(p=2, 10^7, s=0;c=i;pr=p-1;t=0;while(c>0, c=c-1;pr=nextprime(pr+1);s=s+pr; if(c==(i-1)/2, t=pr)); if(s/i==t, print1(t", ");f=1;break)); if(!f, print1("0, ")))

Corrected and extended by Ralf Stephan, Apr 09 2003

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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A055380 Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.

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A082079 Balanced primes of order four.

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A051795 Doubly balanced primes: primes which are averages of both their immediate and their second neighbors.

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A175309 a(n) = the smallest prime prime(k) such that prime(k+j) - prime(k+j-1) = prime(n+k+1-j) - prime(n+k-j) for all j with 1 <= j <= n.

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A081415 Triply balanced primes: primes which are averages of both their immediate neighbor, their second neighbors and their third neighbors.

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