A006562 -id:A006562 - cp's OEIS frontend

A054800 First term of balanced prime quartets: p(m+1)-p(m) = p(m+2)-p(m+1) = p(m+3)-p(m+2).

251, 1741, 3301, 5101, 5381, 6311, 6361, 12641, 13451, 14741, 15791, 15901, 17471, 18211, 19471, 23321, 26171, 30091, 30631, 53611, 56081, 62201, 63691, 71341, 74453, 75521, 76543, 77551, 78791, 80911, 82781, 83431, 84431, 89101, 89381

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

This sequence is infinite if Dickson's conjecture holds. - Charles R Greathouse IV, Apr 23 2011

This is actually the complete list of primes starting a CPAP-4 (set of 4 consecutive primes in arithmetic progression). It equals A033451 for a(1..24), but it contains a(25) = 74453 which starts a CPAP-4 with common difference 18 (the first one with a difference > 6) and therefore is not in A033451. - M. F. Hasler, Oct 26 2018

			a(1) = 251 = prime(54) = A000040(54) and prime(55) - prime(54) = prime(56)-prime(55) = 6. - _Zak Seidov_, Apr 23 2011

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4000 terms from Seidov)

Cf. A006562, A054801 .. A054840.
Cf. A006560 (first prime to start a CPAP-n).
Start of CPAP-4 with given common difference (in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].

Select[Partition[Prime[Range[9000]],4,1],Length[Union[Differences[#]]] == 1&][[All,1]] (* Harvey P. Dale, Aug 08 2017 *)

p=2;q=3;r=5;forprime(s=7,1e4, t=s-r; if(t==r-q&&t==q-p, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Feb 14 2013

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

A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

Original entry on oeis.org

2, 2, 3, 251, 9843019, 121174811

Offset: 1

N. J. A. Sloane

The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n-1). a(6) was found by Lander and Parkin. The next term requires a spacing >= 210. The expected size is a(7) > 10^21 (see link). - Hugo Pfoertner, Jun 25 2004
From Daniel Forgues, Jan 17 2011: (Start)
It is conjectured that there are arithmetic progressions of n consecutive primes for any n.
Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)
a(7) <= 71137654873189893604531, found by P. Zimmermann, cf. J. K. Andersen link. - Bert Dobbelaere, Jul 27 2022

			First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:
a(1) = 2: (2) (degenerate arithmetic progression);
a(2) = 2: (2, 3) (degenerate arithmetic progression);
a(3) = 3: (3, 5, 7);
a(4) = 251: (251, 257, 263, 269);
a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);
a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);

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

Jens Kruse Andersen, The smallest known CPAP-k.
Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.
Daniel Forgues, Wiki about consecutive primes in arithmetic progression.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp., Vol. 21, No. 99 (1967) p 489.
Manfred Toplic, The nine and ten primes project, 2004.
Index entries for sequences related to primes in arithmetic progressions

Cf. A005115, A006562, A093364, A126989.
a(5) corresponds to A052243(20) followed by A052243(21) 9843049.
Cf. A089180: indices primes a(n).
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4), A033451: start of CPAP-4 with common difference 6, A052239: start of first CPAP-4 with common difference 6n.
Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.

Join[{2},Table[SelectFirst[Partition[Prime[Range[691*10^4]],n,1], Length[ Union[ Differences[ #]]] == 1&][[1]],{n,2,6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)

a(n) = A000040(A089180(n)), or A089180(n) = A000720(a(n)). - M. F. Hasler, Oct 27 2018

Edited by Daniel Forgues, Jan 17 2011

A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

Original entry on oeis.org

1, 3, 10, 13, 26, 33, 60, 89, 104, 113, 116, 142, 148, 201, 209, 212, 234, 265, 268, 288, 313, 320, 332, 343, 353, 384, 398, 408, 477, 484, 498, 542, 545, 551, 577, 581, 601, 625, 636, 671, 719, 723, 726, 745, 794, 805, 815, 862, 864, 884, 944, 964, 995, 1054

Offset: 1

Benoit Cloitre, Feb 28 2002

nonn

Also numbers k such that all numbers from prime(k) to prime(k+1) are squarefree. All such primes are twins, so this is a subset of A029707. The other twin primes are A061368. - Gus Wiseman, Dec 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000

A subset of A029707 (lesser index of twin primes).
Prime index of each (prime) term of A061351.
Positions of zeros in A061399.
For perfect power instead of squarefree we have A377436, zeros of A377432.
Positions of zeros in A377784.
The rest of the twin primes are at A378620, indices of A061368.
A000040 lists the primes, differences A001223, (run-lengths A333254, A373821).
A005117 lists the squarefree numbers, differences A076259.
A006562 finds balanced primes.
A013929 lists the nonsquarefree numbers, differences A078147.
A014574 is the intersection of A006093 and A008864.
A038664 locates the first prime gap of size 2n.
A046933 counts composite numbers between primes.
A061398 counts squarefree numbers between primes, zeros A068360.
A120327 gives the least nonsquarefree number >= n.
Cf. A000720, A007674, A013928, A057627, A070321, A071403, A072284, A073247, A112925, A122535, A155752, A224363, A240473, A251092.

Select[Range[100],And@@SquareFreeQ/@Range[Prime[#],Prime[#+1]]&] (* Gus Wiseman, Dec 11 2024 *)

isok(n) = for (k=prime(n)+1, prime(n+1)-1, if (!issquarefree(k), return (0))); 1; \\ Michel Marcus, Apr 29 2016

n such that A061398(n) = prime(n+1)-prime(n)-1.

prime(a(n)) = A061351(n). - Gus Wiseman, Dec 11 2024

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

A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

Original entry on oeis.org

3, 47, 151, 167, 199, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1499, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4397, 4451, 4591, 4651, 4679, 4987, 5101, 5107, 5297, 5381, 5387

Offset: 1

Miklos Kristof, Sep 18 2006

nonn

Subsets are A047948, A052188, A052189, A052190, A052195, A052197, A052198, etc. - R. J. Mathar, Apr 11 2008
Could be generated by searching for cases A001223(i) = A001223(i+1), writing down A000040(i). - R. J. Mathar, Dec 20 2008
a(n) = A006562(n) - A117217(n). - Zak Seidov, Feb 12 2013
These are primes for which the subsequent prime gaps are equal, so (p(k+2)-p(k+1))/(p(k+1)-p(k)) = 1. It is conjectured that prime gaps ratios equal to one are less frequent than those equal to 1/2, 2, 3/2, 2/3, 1/3 and 3. - Andres Cicuttin, Nov 07 2016

			The prime 7 is not in the list, because in the triple (7, 11, 13) of successive primes, 11 is not equal (7 + 13)/2 = 10.
The second term, 47, is the first prime in the triple (47, 53, 59) of primes, where 53 is the mean of 47 and 59.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Cf. A006562, A062839, A102552, A117217, A181424.

a122535 = a000040 . a064113  -- Reinhard Zumkeller, Jan 20 2012

Clear[d2, d1, k]; d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] == 0, Prime[n], {}], {n, 1, 1000}]] (* Roger L. Bagula, Nov 13 2008 *)
Transpose[Select[Partition[Prime[Range[750]], 3, 1], #[[2]] == (#[[1]] + #[[3]])/2 &]][[1]]  (* Harvey P. Dale, Jan 09 2011 *)

A122535()={n=3;ctr=0;while(ctr<50, avgg=( prime(n-2)+prime(n) )/2;
if( prime(n-1) ==avgg, ctr+=1;print( ctr,"  ",prime(n-2) )  );n+=1); } \\ Bill McEachen, Jan 19 2015

{A000040(i): A000040(i+1)= (A000040(i)+A000040(i+2))/2 }. - R. J. Mathar, Dec 20 2008

a(n) = A000040(A064113(n)). - Reinhard Zumkeller, Jan 20 2012

More terms from Roger L. Bagula, Nov 13 2008

Definition rephrased by R. J. Mathar, Dec 20 2008

A082077 Balanced primes of order two.

Original entry on oeis.org

79, 281, 349, 439, 643, 677, 787, 1171, 1733, 1811, 2141, 2347, 2389, 2767, 2791, 3323, 3329, 3529, 3929, 4157, 4349, 4751, 4799, 4919, 4951, 5003, 5189, 5323, 5347, 5521, 5857, 5861, 6287, 6337, 6473, 6967, 6997, 7507, 7933, 8233, 8377, 8429, 9377, 9623, 9629, 10093, 10333

Offset: 1

Labos Elemer, Apr 08 2003

nonn

The arithmetic mean of 4 primes in its "neighborhood"; not to be confused with 'Doubly balanced primes' (A051795).
Balanced primes of order two are not necessarily balanced of order one (A006562) or three (A082078).
Subsequence of A219478, Peter Schorn, May 01 2025

			p = 79 = (71 + 73 + 79 + 83 + 89)/5 = 395/5 i.e. it is both the arithmetic mean and median.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006562, A082078, A082079, A096697, A096698, A096699, A096700, A096701, A096702, A096703, A096704, A096693, A082080, A081415, A051795, A006562, A219478.

Do[s3=Prime[n]+Prime[n+1]+Prime[n+2]; s5=Prime[n-1]+s3+Prime[n+3]; If[Equal[s5/5, Prime[n+1]], Print[Prime[n+1]]], {n, 3, 3000}]
Select[Partition[Prime[Range[1500]],5,1],Mean[#]==#[[3]]&][[All,3]] (* Harvey P. Dale, Nov 04 2019 *)

p=2;q=3;r=5;s=7;forprime(t=11,1e9,if(p+q+s+t==4*r,print1(r", ")); p=q; q=r; r=s; s=t) \\ Charles R Greathouse IV, Nov 20 2012

A054803 Fourth term of balanced prime quartets: p(m-2)-p(m-3) = p(m-1)-p(m-2) = p(m)-p(m-1).

Original entry on oeis.org

269, 1759, 3319, 5119, 5399, 6329, 6379, 12659, 13469, 14759, 15809, 15919, 17489, 18229, 19489, 23339, 26189, 30109, 30649, 53629, 56099, 62219, 63709, 71359, 74507, 75539, 76597, 77569, 78809, 80929, 82799, 83449, 84449, 89119, 89399

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A006562, A054800-A054840, A181424.

Transpose[Select[Partition[Prime[Range[9000]],4,1],Length[Union[ Differences[ #]]]==1&]][[4]] (* Harvey P. Dale, Aug 27 2012 *)

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

cp's OEIS Frontend

A054800 First term of balanced prime quartets: p(m+1)-p(m) = p(m+2)-p(m+1) = p(m+3)-p(m+2).

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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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A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

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A068361 Numbers n such that the number of squarefree numbers between prime(n) and prime(n+1) = prime(n+1)-prime(n)-1.

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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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A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.