A051635 -id:A051635 - cp's OEIS frontend

A124661 Primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for k = 1..n-2.

2, 3, 5, 7, 13, 19, 23, 31, 43, 47, 73, 83, 109, 113, 181, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789

Offset: 1

Artur Jasinski, Dec 23 2006

The first two primes, 2 and 3, are tested against an empty set of k, and we include them, defining such a test to have a positive outcome.
McNew's "popular primes" sequence (A385503) has the same first 14 terms, differing first by excluding 181. McNew says that a prime p is "popular" on an interval [2, k] if no prime occurs more frequently than p as the greatest prime factor (gpf, A006530) of the integers in that interval. - N. J. A. Sloane, Jul 25 2017 and Peter Munn, Jul 01 2025
See the Pomerance link for a proof that the sequence is infinite. - Peter Munn, Jul 01 2025

			prime(11)=31 is in the sequence because prime(10)+prime(12) = 66, prime(9)+prime(13) = 64,..., prime(2)+prime(20) = 74 are all >= 62 = 2*31.
prime(10) = 29 is not in the sequence because prime(9)+prime(11) = 54 for example is smaller than 58 = 2*29.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Nathan McNew, Popular values of the largest prime divisor function, arXiv:1504.05985 [math.NT], 2015.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
C. Pomerance, The prime number graph, Math. Comp. 33 (1979) 399--408. - _Nathan McNew_, Apr 04 2014

Cf. A006530, A051635, A385503.

Select[Prime@ Range@ 300, Function[{p, n}, NoneTrue[Range[n - 2], Prime[n - #] + Prime[n + #] < 2 p &]] @@ {#, PrimePi@ #} &] (* Michael De Vlieger, Jul 25 2017 *)

isok(p) = {n = primepi(p); for (k=1, n-2, if (prime(n-k) + prime(n+k) < 2*p, return (0));); return (1);}
lista(nn) = {for(n=1, nn, if (isok(prime(n)), print1(prime(n), ", ");););} \\ Michel Marcus, Nov 03 2013

from sympy import prime
A124661_list = []
for n in range(1,10**6):
    p = prime(n)
    for k in range(1,n-1):
        if prime(n-k)+prime(n+k) < 2*p:
            break
    else:
        A124661_list.append(p) # Chai Wah Wu, Jul 25 2017

Sequence extended by R. J. Mathar, Mar 28 2010

Edited, restoring previous name, by Peter Munn, Jul 01 2025

A054806 Third term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Original entry on oeis.org

41, 71, 101, 227, 281, 311, 461, 487, 617, 641, 727, 757, 857, 881, 937, 1091, 1301, 1427, 1451, 1481, 1549, 1663, 1667, 1697, 1741, 1783, 1787, 1861, 1867, 1871, 1993, 1997, 2063, 2141, 2339, 2377, 2381, 2467, 2473, 2521, 2531, 2539, 2591, 2617, 2657

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..10000, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Select[Partition[Prime[Range[400]],4,1],Max[Differences[#,2]]<0&][[All,3]] (* Harvey P. Dale, Aug 28 2021 *)

a(n) = nextprime(A054805(n)) = prevprime(A054807(n)), nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Offset corrected to 1 by M. F. Hasler, Oct 27 2018

Definition clarified by N. J. A. Sloane, Aug 28 2021

A054808 First term of strong prime quintets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

Original entry on oeis.org

1637, 1759, 1831, 1847, 1979, 2357, 2447, 2477, 2503, 3413, 3433, 4177, 4493, 5237, 5399, 5419, 6011, 6619, 7219, 7253, 7727, 7853, 7907, 8123, 8467, 9551, 9587, 11003, 11353, 11551, 11813, 12379, 13841, 14797, 15107, 15511, 16007, 16273, 16787, 16993, 17359, 18149, 18289

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

First member of pairs of consecutive primes in A054804 (first of strong quartets): The first 10^4 terms of that sequence yield over 2000 terms of this sequence. - M. F. Hasler, Oct 27 2018

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quartets (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartets, quintets, sextets; A054819 .. A054840: members of weak prime quartets, quintets, sextets, septets.

okQ[l_]:=Module[{d=Differences[l]},d[[1]]>d[[2]]>d[[3]]>d[[4]]]; Transpose[ Select[Partition[Prime[Range[2000]],5,1],okQ]][[1]] (* Harvey P. Dale, Aug 15 2011 *)

A054808=List();for(i=2,1e4,A054804[i]==A054805[i-1]&&listput(A054808,A054804[i-1])) \\ M. F. Hasler, Oct 27 2018

a(n) = prevprime(A054809(n)); A054808 = {m = A054804(n) | nextprime(m) = A054804(n+1)}; nextprime = A151800, prevprime = A151799. - M. F. Hasler, Oct 27 2018

Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054835 Second term of weak prime septet: p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3) < p(m+5)-p(m+4).

Original entry on oeis.org

15377, 64921, 68209, 68899, 128983, 128987, 143513, 154081, 158003, 192377, 221719, 222389, 244463, 249727, 285289, 318679, 337279, 354373, 357829, 374177, 385393, 394729, 402583, 402587, 419599, 439163, 441913, 448379, 457399, 457673, 458191, 482509, 527983, 529813, 577531, 582763, 655913

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..1000, Oct 27 2018

Cf. A051635, A229832.

Cf. A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

a(1) = A229832(5). - Jonathan Sondow, Oct 13 2013

a(n) = A151800(A054834(n)) = A151799(A054836(n)), A151800 = nextprime, A151799 = prevprime; A054835 = { m = A054828(n) | m = nextprime(A054828(n-1)) }. - M. F. Hasler, Oct 27 2018

More terms from M. F. Hasler, Oct 27 2018

A054838 Fifth term of weak prime septet: p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1).

Original entry on oeis.org

15401, 64951, 68227, 68917, 129001, 129011, 143537, 154111, 158029, 192407, 221737, 222437, 244493, 249763, 285343, 318701, 337301, 354391, 357883, 374219, 385417, 394747, 402601, 402613, 419623, 439199, 441953, 448421, 457421, 457697, 458219, 482527, 528001

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051635; A054800 .. A054803: members of balanced prime quartets (= consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime quartet, quintet, sextet; A054819 .. A054840: members of weak prime quartet, quintet, sextet, septets.

Select[Partition[Prime[Range[7000]],7,1],Min[Differences[#,2]]>0&][[All,5]] (* Harvey P. Dale, Oct 15 2016 *)

a(n) = A151800(A054837(n)) = A151799(A054839(n)), A151800 = nextprime, A151799 = prevprime; A054838 = { m = A054831(n) | m = nextprime(A054831(n-1)) }. - M. F. Hasler, Oct 27 2018

More terms from Harvey P. Dale, Oct 15 2016

A068828 Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).

Original entry on oeis.org

2, 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

Offset: 1

Amarnath Murthy, Mar 08 2002

Or, bad primes (version 1): primes not in A046869. - Jonathan Vos Post, Aug 20 2007
The idea can be extended by defining a geometrically weak prime of order k to be a prime which is less than the geometric mean of r neighbors on both sides for all r = 1 to k and not true for r = k+1. A similar extension could be defined for the sequence A051635.
It is easy to show that, except for the twin prime pair (3,5), the larger prime of every twin prime pair is in this sequence. The smaller prime of the pair is always in A046869. - T. D. Noe, Feb 19 2008

			23 belongs to this sequence as 23^2 = 529 < 19*29 = 551.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A051634, A051635, A006562, A000040, A046869.

Join[{2}, Prime[Select[Range[2, 120], Prime[ # ]^2 <= Prime[ # - 1]*Prime[ # + 1]&]]] (* Stefan Steinerberger, Aug 21 2007 *)
Join[{2},Transpose[Select[Partition[Prime[Range[500]],3,1],#[[2]]< GeometricMean[ {#[[1]],#[[3]]}]&]][[2]]] (* Harvey P. Dale, Apr 05 2014 *)

prime(k)^2 <= prime(k-1)*prime(k+1).

Corrected and extended by Stefan Steinerberger, Aug 21 2007

Edited by N. J. A. Sloane, Feb 19 2008

A264720 Numbers that are less than the average of their closest flanking primes.

Original entry on oeis.org

3, 7, 8, 13, 14, 19, 20, 23, 24, 25, 31, 32, 33, 38, 43, 44, 47, 48, 49, 54, 55, 61, 62, 63, 68, 73, 74, 75, 80, 83, 84, 85, 89, 90, 91, 92, 98, 103, 104, 109, 110, 113, 114, 115, 116, 117, 118, 119, 128, 131, 132, 133, 139, 140, 141, 142, 143, 151, 152, 153, 158

Offset: 1

Chris Boyd, Nov 21 2015

Numbers that are nearer to the immediately previous prime than to the immediately next prime.
This sequence may be viewed as a generalization of A051635 (the weak primes) that includes qualifying composite numbers.
The union of this sequence with A264719 & A145025 is A000027 (omitting 1 & 2).

			a(11) = 31 because 31 < (29 + 37)/2 = 33.
a(12) = 32 because 32 < (31 + 37)/2 = 34.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A051634, A145025, A264719, A264721, A264722.

Select[Range@ 162, # < (NextPrime[#, -1] + NextPrime@ #)/2 &] (* Michael De Vlieger, Nov 22 2015 *)

test(n)= {if(n-precprime(n-1)2,return(1),return(0))}
for(i=1,200,if(test(i),print1(i,", ")))

A054809 Second term of strong prime 5-tuples: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

Original entry on oeis.org

1657, 1777, 1847, 1861, 1987, 2371, 2459, 2503, 2521, 3433, 3449, 4201, 4507, 5261, 5407, 5431, 6029, 6637, 7229, 7283, 7741, 7867, 7919, 8147, 8501, 9587, 9601, 11027, 11369, 11579, 11821, 12391, 13859, 14813, 15121, 15527, 16033, 16301, 16811, 17011, 17377

Offset: 1

Henry Bottomley, Apr 10 2000

Initial member of pairs of consecutive primes in A054805 (second of quadruples): The first 10^4 terms of that sequence yield over 2000 terms of this sequence. - M. F. Hasler, Oct 27 2018

M. F. Hasler, Table of n, a(n) for n = 1..2000

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quadruples (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime 4-tuples, 5-tuples, 6-tuples; A054819 .. A054840: members of weak prime 4-tuples, ..., 7-tuples.

spqQ[n_]:=Module[{difs=Differences[n]},difs[[1]]>difs[[2]]> difs[[3]]> difs[[4]]]; Transpose[Select[Partition[Prime[ Range[2000]],5,1], spqQ]][[2]] (* Harvey P. Dale, May 06 2012 *)

a(n) = nextprime(A054808(n)) = prevprime(A054810(n)), nextprime = A151800, prevprime = A151799; A054809 = {m = A054805(n) | nextprime(m) = A054805(n+1)}. - M. F. Hasler, Oct 27 2018

Corrected by Harvey P. Dale, May 06 2012

Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054810 Third term of strong prime 5-tuples: p(m-1)-p(m-2) > p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1).

Original entry on oeis.org

1663, 1783, 1861, 1867, 1993, 2377, 2467, 2521, 2531, 3449, 3457, 4211, 4513, 5273, 5413, 5437, 6037, 6653, 7237, 7297, 7753, 7873, 7927, 8161, 8513, 9601, 9613, 11047, 11383, 11587, 11827, 12401, 13873, 14821, 15131, 15541, 16057, 16319

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

M. F. Hasler, Table of n, a(n) for n = 1..2000 (first 1001 terms by Harvey P. Dale).

Cf. A051634, A051635; A054800 .. A054803: members of balanced prime quadruples (= 4 consecutive primes in arithmetic progression); A054804 .. A054818: members of strong prime 4-tuples, 5-tuples, 6-tuples; A054819 .. A054840: members of weak prime 4-tuples, ..., 7-tuples.

spqQ[{a_,b_,c_,d_,e_}]:=(b-a)>(c-b)>(d-c)>(e-d); Transpose[ Select[ Partition[ Prime[ Range[2000]],5,1],spqQ]][[3]] (* Harvey P. Dale, Feb 25 2013 *)

Edited and offset corrected to 1 by M. F. Hasler, Oct 27 2018

A054828 First term of weak prime sextet: p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3) < p(m+5)-p(m+4).

Original entry on oeis.org

2903, 13463, 13901, 14947, 15373, 15377, 21397, 21557, 21859, 28277, 30869, 33199, 35591, 37691, 42221, 42569, 45821, 55661, 64661, 64919, 64921, 68207, 68209, 68897, 68899, 73939, 74201, 78577, 83089, 85513, 87313, 88001, 90907

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051635, A054800-A054840.

Transpose[Select[Partition[Prime[Range[9000]],6,1],AllTrue[ Differences[ #,2], Positive]&]] [[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 12 2014 *)

cp's OEIS Frontend

A124661 Primes prime(n) such that prime(n-k)+prime(n+k) >= 2*prime(n) for k = 1..n-2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A054806 Third term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A054808 First term of strong prime quintets: p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2) > p(m+4)-p(m+3).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A054835 Second term of weak prime septet: p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1) < p(m+3)-p(m+2) < p(m+4)-p(m+3) < p(m+5)-p(m+4).

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A054838 Fifth term of weak prime septet: p(m-3)-p(m-4) < p(m-2)-p(m-3) < p(m-1)-p(m-2) < p(m)-p(m-1) < p(m+1)-p(m) < p(m+2)-p(m+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A068828 Geometrically weak primes: primes that are smaller than the geometric mean of their neighbors (2 is included by convention).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A264720 Numbers that are less than the average of their closest flanking primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A054809 Second term of strong prime 5-tuples: p(m)-p(m-1) > p(m+1)-p(m) > p(m+2)-p(m+1) > p(m+3)-p(m+2).

Views

Author

Keywords