A051634 -id:A051634 - cp's OEIS frontend

A178943 Primes that are not balanced primes (see A006562).

2, 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 269, 271, 277, 281, 283, 293

Offset: 1

Paul Muljadi, Dec 30 2010

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement with respect to A000040 gives A006562.

Union of {2}, A051634 and A051634; A225496 (multiplicative closure).

a178943 n = a178943_list !! (n-1)
a178943_list = 2 : 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

Transpose[ Select[ Partition[ Prime@ Range@ 65, 3, 1],#[[2]]!=(#[[1]]+#[[3]])/2&]][[2]]

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

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

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

A264719 Numbers that are greater than the average of their closest flanking primes.

Original entry on oeis.org

10, 11, 16, 17, 22, 27, 28, 29, 35, 36, 37, 40, 41, 46, 51, 52, 57, 58, 59, 65, 66, 67, 70, 71, 77, 78, 79, 82, 87, 88, 94, 95, 96, 97, 100, 101, 106, 107, 112, 121, 122, 123, 124, 125, 126, 127, 130, 135, 136, 137, 145, 146, 147, 148, 149, 155, 156, 161, 162

Offset: 1

Chris Boyd, Nov 21 2015

Numbers that are nearer to the immediately next prime than to the immediately previous prime.
This sequence may be viewed as a generalization of A051634 (the strong primes) that includes qualifying composite numbers.
The union of this sequence with A264720 & A145025 is A000027 (omitting 1 & 2).

			a(11) = 37 because 37 > (31 + 41)/2 = 36.
a(12) = 40 because 40 > (37 + 41)/2 = 37.

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

Cf. A051634, A145025, A264720, A264721, A264722.

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

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

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

A102552 a(n) = prime(n) - (prime(n+1) + prime(n-1))/2.

Original entry on oeis.org

0, -1, 1, -1, 1, -1, -1, 2, -2, 1, 1, -1, -1, 0, 2, -2, 1, 1, -2, 1, -1, -1, 2, 1, -1, 1, -1, -5, 5, -1, 2, -4, 4, -2, 0, 1, -1, 0, 2, -4, 4, -1, 1, -5, 0, 4, 1, -1, -1, 2, -4, 2, 0, 0, 2, -2, 1, 1, -4, -2, 5, 1, -1, -5, 4, -2, 4, -1, -1, -1, 1, 0, 1, -1, -1, 2, -2, -1, 4, -4, 4, -2, 1, -1, -1, 2, 1, -1, -4, 2, 2, -2, 2, -1, -3, 5, -8, 6, -2, 2, 0, 2, -2

Offset: 3

Yasutoshi Kohmoto, Feb 25 2005

sign

			a(6)=-1 because 13-(17+11)/2=-1.

Eric Weisstein, CRC Concise Encyclopedia of Mathematics, 1998, page 1321.

G. C. Greubel, Table of n, a(n) for n = 3..10000

Cf. A000040, A006562, A036263, A051634, A051635, A066875.

A102552:= func< n | (2*NthPrime(n)-NthPrime(n+1)-NthPrime(n-1))/2 >;
[A102552(n): n in [3..120]]; // G. C. Greubel, Feb 02 2025

a:=n->ithprime(n)-(ithprime(n+1)+ithprime(n-1))/2: seq(a(n),n=3..95); # Emeric Deutsch, Mar 02 2005

f[n_] := Prime[n] - (Prime[n - 1] + Prime[n + 1])/2; Table[f[n], {n, 3, 107}] (* Robert G. Wilson v, Sep 25 2006 *)
#[[2]]-(#[[1]]+#[[3]])/2&/@Partition[Prime[Range[2,110]],3,1] (* Harvey P. Dale, Sep 21 2013 *)

a(n) = prime(n)-(prime(n+1)+prime(n-1))/2;
vector(100,n,a(n+2)) \\ Joerg Arndt, Jan 20 2015

from sympy import sieve as p
def A102552(n): return p[n]-(p[n+1]+p[n-1])//2 # Karl-Heinz Hofmann, May 22 2024

a(n) = (1/2)*(A001223(n) - A001223(n+1)).

a(n) = -A036263(n-1)/2. - T. D. Noe, Oct 06 2006 [corrected by Georg Fischer, Oct 19 2023]

More terms from Emeric Deutsch, Mar 02 2005

cp's OEIS Frontend

A178943 Primes that are not balanced primes (see A006562).

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

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

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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).

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

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A264719 Numbers that are greater than the average of their closest flanking primes.

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

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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).

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