A181424 -id:A181424 - 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

A064113 Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.

Original entry on oeis.org

2, 15, 36, 39, 46, 54, 55, 73, 102, 107, 110, 118, 129, 160, 164, 184, 187, 194, 199, 218, 239, 271, 272, 291, 339, 358, 387, 419, 426, 464, 465, 508, 520, 553, 599, 605, 621, 629, 633, 667, 682, 683, 702, 709, 710, 733, 761, 791, 813, 821, 822, 829, 830

Offset: 1

Jason Earls, Sep 08 2001

n such that d(n) = d(n+1), where d(n) = prime(n+1) - prime(n) = A001223(n).
Of interest because when I generalize it to d(n) = d(n+2), d(n) = d(n+3), etc. I am unable to find any positive number k such that d(n) = d(n+k) has no solution.
From Lei Zhou, Dec 06 2005: (Start)
When (1/3)*(prime(k) + prime(k+1) + prime(k+2)) is prime, then it is equal to prime(k+1).
Also, indices k such that (prime(k)+prime(k+2))/2 = prime(k+1).
The Mathematica program is based on the alternative definition. (End)
Inflection and undulation points of the primes, i.e., positions of zeros in A036263, the second differences of the primes. - Gus Wiseman, Mar 24 2020

			a(2) = 15 because (p(15)+p(16)+p(17)) = 1/3(47 + 53 + 59) = 53 (prime average of three successive primes).
Splitting the prime gaps into anti-runs gives: (1,2), (2,4,2,4,2,4,6,2,6,4,2,4,6), (6,2,6,4,2,6,4,6,8,4,2,4,2,4,14,4,6,2,10,2,6), (6,4,6), ... Then a(n) is the n-th partial sum of the lengths of these anti-runs. - _Gus Wiseman_, Mar 24 2020

Harry J. Smith, Table of n, a(n) for n=1..1000
Wikipedia, Inflection point

Indices of zeros in A036263 (second differences of primes).
Indices (A000720 = primepi) of balanced primes A006562, minus 1.
Cf. A001223, A024675, A075540, A075541.
Cf. A262138.
Complement of A333214.
First differences are A333216.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for weak ascents is A333230.
The version for weak descents is A333231.
A triangle for anti-runs of compositions is A106356.
Lengths of maximal runs of prime gaps are A333254.
Anti-runs of compositions in standard order are A333381.
Cf. A000040, A084758, A124762, A124767, A238279.

import Data.List (elemIndices)
a064113 n = a064113_list !! (n-1)
a064113_list = map (+ 1) $ elemIndices 0 a036263_list
-- Reinhard Zumkeller, Jan 20 2012

ct = 0; Do[If[(Prime[k] + Prime[k + 2] - 2*Prime[k + 1]) == 0, ct++; n[ct] = k], {k, 1, 2000}]; Table[n[k], {k, 1, ct}] (* Lei Zhou, Dec 06 2005 *)
Join@@Position[Differences[Array[Prime,100],2],0] (* Gus Wiseman, Mar 24 2020 *)

d(n) = prime(n+1)-prime(n); j=[]; for(n=1,1500, if(d(n)==d(n+1), j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (d(m)==d(m+1), write("b064113.txt", n++, " ", m); if (n==1000, break)) ) } \\ Using d(n) above. - Harry J. Smith, Sep 07 2009

[n | n<-[1..888], !A036263(n)] \\ M. F. Hasler, Oct 15 2024

\\ More efficient for larges range of n:
A064113_upto(N, n=1, L=List(), q=prime(n+1), d=q-prime(n))={forprime(p=1+q,, if(d==d=p-q, listput(L,n); #LM. F. Hasler, Oct 15 2024

from itertools import count, islice
from sympy import prime, nextprime
def A064113_gen(startvalue=1): # generator of terms >= startvalue
    c = max(startvalue,1)
    p = prime(c)
    q = nextprime(p)
    r = nextprime(q)
    for k in count(c):
        if p+r==(q<<1):
            yield k
        p, q, r = q, r, nextprime(r)
A064113_list = list(islice(A064113_gen(),20)) # Chai Wah Wu, Feb 27 2024

A036263(a(n)) = 0; A122535(n) = A000040(a(n)); A006562(n) = A000040(a(n) + 1); A181424(n) = A000040(a(n) + 2). - Reinhard Zumkeller, Jan 20 2012
A262138(2*a(n)) = 0. - Reinhard Zumkeller, Sep 12 2015
a(n) = A000720(A006562(n)) - 1, where A000720 = (prime)pi, A006562 = balanced primes. - M. F. Hasler, Oct 15 2024

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

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 *)

A358528 a(n) = n-th prime p(k) such that p(k) - p(k-1) > p(k-1) - p(k-2).

Original entry on oeis.org

5, 11, 17, 23, 29, 37, 47, 53, 67, 79, 89, 97, 107, 113, 127, 137, 149, 157, 173, 191, 197, 211, 233, 239, 251, 277, 293, 307, 317, 331, 347, 353, 359, 367, 389, 397, 409, 419, 431, 439, 449, 457, 467, 479, 499, 509, 521, 541, 557, 577, 587, 607, 631, 647

Offset: 1

Clark Kimberling, Nov 21 2022

This sequence, together with A358530 and A181424, partition the set of primes >= 5. The corresponding sequences of indices, A358529, A358531, and A356347, partition the set of positive integers >= 3.

			   n      1   2   3   4   5   6   7
   k      3   5   7   9  10  12  15
   p(n)   5  11  17  23  29  37  47

Cf. A358529, A358530, A358531, A181424, A356347.

t = Select[2 + Range[140],
Prime[#] - Prime[# - 1] > Prime[# - 1] - Prime[# - 2] &]  (* A358529 *)
Prime[t]  (* A358528 *)

A358529 Indices of the primes in A358528.

Original entry on oeis.org

3, 5, 7, 9, 10, 12, 15, 16, 19, 22, 24, 25, 28, 30, 31, 33, 35, 37, 40, 43, 45, 47, 51, 52, 54, 59, 62, 63, 66, 67, 69, 71, 72, 73, 77, 78, 80, 81, 83, 85, 87, 88, 91, 92, 95, 97, 98, 100, 102, 106, 107, 111, 115, 118, 119, 122, 124, 125, 126, 128, 133, 136

Offset: 1

Clark Kimberling, Nov 21 2022

This sequence, together with A358531 and A356347, partition the set of positive integers >= 3.

			  n      1   2   3   4   5   6   7
  k      3   5   7   9  10  12  15
  p(n)   5  11  17  23  29  37  47

Cf. A358528, A358530, A358531, A181424, A356347.

t = Select[2 + Range[140],
Prime[#] - Prime[# - 1] > Prime[# - 1] - Prime[# - 2] &]  (* A358529 *)
Prime[t]  (* A358528 *)

a(n) = A233671(n) + 1.

A358530 a(n) = n-th prime prime(k) such that prime(k) - prime(k-1) < prime(k-1) - prime(k-2).

Original entry on oeis.org

13, 19, 31, 41, 43, 61, 71, 73, 83, 101, 103, 109, 131, 139, 151, 167, 181, 193, 199, 227, 229, 241, 257, 271, 281, 283, 311, 313, 337, 349, 373, 383, 401, 421, 433, 443, 461, 463, 487, 491, 503, 523, 547, 563, 571, 593, 601, 617, 619, 641, 643, 661, 677

Offset: 1

Clark Kimberling, Nov 21 2022

This sequence, together with A358528 and A181424, partition the set of primes >= 5. The corresponding sequences of indices, A358531, A358529, and A356347, partition the set of positive integers >= 3.

			  n           1   2   3   4   5   6   7
  k           6   8  11  13  14  18  20
  prime(n)   13  19  31  41  43  61  71

Cf. A001223, A051634, A079419, A358528, A358529, A358531, A181424, A356347.

t = Select[2 + Range[140],
Prime[#] - Prime[# - 1] < Prime[# - 1] - Prime[# - 2] &]  (* A358531 *)
Prime[t]  (* A358530 *)

a(n) = A151800(A051634(n)). - Andrew Howroyd, Sep 21 2024

Incorrect formula removed by Georg Fischer, Sep 21 2024

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A356347 Indices of the primes in A181424.

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

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A064113 Indices k such that (1/3)*(prime(k)+prime(k+1)+prime(k+2)) is a prime.

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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.

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

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A358528 a(n) = n-th prime p(k) such that p(k) - p(k-1) > p(k-1) - p(k-2).

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A358529 Indices of the primes in A358528.