A006562 -id:A006562 - cp's OEIS frontend

A053711 Numbers d such that, for some k, the upper and lower primes closest to k! are k! + d and k! - d.

1, 7, 11, 31, 397, 409, 1657, 2131, 7607

Offset: 1

Labos Elemer, Feb 10 2000

This sequence lists d = nextprime(k!) - k! = prevprime(k!) - k! for k in A053709.

			For k = 10, k! = 3628800, d = 11, and the closest primes to 10! are q = 10! - 11 = 3628789 and p = 10! + 11 = 3628811. The differences d are listed here.

Cf. A033932, A033933, A006990, A037151, A006562, A053709, A053710.

Reap[Do[If[SameQ @@ #, Sow@ First[#]] &@ Abs[# - NextPrime[#, {-1, 1}]] &[i!], {i, 200}]][[-1, -1]] (* Michael De Vlieger, Jan 14 2022 *)

a(5)-a(8) from Donovan Johnson, Oct 12 2008

a(9) from Hans Havermann, Aug 15 2014

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

A085909 Smallest prime p>prime(n) such that p+prime(n+1)-prime(n) is the next prime after p; or a(n)=0 if no such prime exists.

Original entry on oeis.org

0, 5, 11, 13, 17, 19, 29, 37, 31, 41, 47, 43, 59, 67, 53, 61, 71, 73, 79, 101, 83, 97, 131, 359, 103, 107, 109, 137, 127, 293, 163, 151, 149, 181, 179, 157, 167, 193, 173, 233, 191, 241, 197, 223, 227, 211, 467, 229, 239, 277, 251, 269, 283, 257, 263, 271, 281

Offset: 1

Amarnath Murthy, Jul 09 2003

nonn

A001223(n) = A001223(A049084(a(n))); a(A001359(n)) = A001359(n+1); conjecture: a(n) > 0 for n > 1 (implies the twin prime conjecture). - Reinhard Zumkeller, Jan 26 2004

For n > 1, a(n) >= prime(n+1) and a(n) = prime(n+1) if prime(n+1) is a balanced prime (A006562). - Zak Seidov, Jun 03 2015

Zak Seidov, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Difference Function

Cf. A085910, A001223, A049084, A001359, A006562.

P = primes(5000); A = zeros(1, length(P));D = P(2:end) - P(1:(length(P) - 1)); for i = 2:2:(max(D));f = find(D == i); A(f(1:(length(f) - 1))) = P(f(2:end));end;A(2:100) % David Wasserman, Jan 26 2004

a[1] = 0; a[n_] := For[p = Prime[n+1]; d = p - Prime[n], True, p = q, q = NextPrime[p]; If[d == q - p, Return[p]]]; (* Jean-François Alcover, Feb 24 2015 *)

More terms from Reinhard Zumkeller and David Wasserman, Jan 26 2004

Edited by N. J. A. Sloane, Oct 21 2008 at the suggestion of R. J. Mathar

A126557 Primes in A126556.

Original entry on oeis.org

174737, 224327, 433813, 541447, 787243, 969667, 980081, 1080787, 1286581, 1372979, 1534513, 1567037, 1570649, 1577189, 1659673, 1726993, 1931291, 2242883, 2282041, 2415557, 2460827, 3162503, 3711047, 4090787, 4450373

Offset: 1

Artur Jasinski, Dec 27 2006

nonn

Prime interprimes of third order.

Primes that are the arithmetic mean of two consecutive prime interprimes of second order; primes of the form (A126555(k)+A126555(k+1))/2.

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

Cf. A006562 (balanced primes), A024675 (interprimes), A126554 (interprimes of second order), A126555 (prime interprimes of second order), A126556 (interprimes of third order).

{m=5000000;a=0;g=0;p=2;q=3;r=5;while(r<=m,if((p+r)/2==q,if(a>0,b=(a+q)/2;if(isprime(b),if(g>0,if(isprime(h=(g+b)/2),print1(h,",")));g=b));a=q);p=q;q=r; r=nextprime(r+1))} \\ Klaus Brockhaus, Jan 11 2007

Edited and extended by Klaus Brockhaus, Jan 11 2007

A178954 Primes prime(j) which cannot be written as 2*prime(j) = prime(j+k) + prime(j-k) for any 0 < k < j.

Original entry on oeis.org

2, 3, 7, 19, 23, 43, 47, 73, 79, 109, 113, 149, 163, 199, 223, 227, 229, 239, 241, 269, 271, 281, 283, 293, 313, 317, 463, 467, 499, 503, 509, 523, 619, 659, 661, 673, 677, 683, 691, 719, 829, 839, 859, 883, 887, 967, 1049, 1063, 1069, 1109, 1117, 1129, 1153, 1163, 1201

Offset: 1

Juri-Stepan Gerasimov, Jan 05 2011

nonn

Sequence A127925, in which 2*prime(j) < prime(j+k) + prime(j-k) for all 0 < k < j, is a subsequence of this sequence. According to section A14 of Guy, Pomerance proved that A127925 is an infinite sequence. Hence, this sequence is also infinite. - T. D. Noe, Jan 10 2011

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer, 2004.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A100484, A006562, A178609, A178953, A178970.

Cf. A178670.

A178609 := proc(n) for k from n-1 to 0 by -1 do if ithprime(n-k)+ithprime(n+k)=2*ithprime(n) then return k; end if; end do: end proc:
for n from 1 to 200 do if A178609(n) = 0 then printf("%d,",ithprime(n)) ; end if; end do: # R. J. Mathar, Jan 05 2011

From R. J. Mathar, Jan 05 2011: (Start)
{A000040(k): A178609(k)=0}.
a(n) = A000040(A178953(n)). (End)

A184247 Primes, q, such that for three consecutive primes, p, q & r, with p

Original entry on oeis.org

5, 11, 17, 29, 41, 53, 59, 71, 97, 101, 107, 137, 149, 157, 173, 179, 191, 197, 211, 223, 227, 239, 257, 263, 269, 281, 311, 347, 373, 397, 419, 431, 457, 461, 487, 499, 521, 541, 563, 569, 593, 599, 607, 617, 641, 653, 659, 673, 733, 769, 809, 821, 827, 857

Offset: 1

Robert G. Wilson v, Jan 10 2011

The distance between the cited prime above to its immediate predecessor is divisible by the distance from that prime to its immediate successor.

Intersection(A184247, A184248): 5, 53, 157, 173, 211, ..., = A006562: Balanced primes (of order 1).

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

Cf. A184248.

fQ[n_] := Block[{p = NextPrime[n, -1], q = n, r = NextPrime[n]}, IntegerQ[(q - p)/(r - q)]]; Select[ Prime@ Range[2, 50], fQ]
Select[Partition[Prime[Range[150]],3,1],IntegerQ[(#[[2]]-#[[1]])/(#[[3]]- #[[2]])]&][[All,2]] (* Harvey P. Dale, Jul 26 2018 *)

A293395 The initial member of 5 consecutive primes whose arithmetic mean is the middle member.

Original entry on oeis.org

71, 271, 337, 431, 631, 661, 769, 1153, 1721, 1789, 2131, 2339, 2381, 2749, 2777, 3313, 3319, 3517, 3919, 4139, 4337, 4729, 4789, 4903, 4937, 4993, 5171, 5303, 5323, 5507, 5849, 5851, 6271, 6323, 6451, 6959, 6983, 7489, 7919, 8221, 8363, 8419, 9349, 9613, 9619

Offset: 1

K. D. Bajpai, Oct 08 2017

nonn

3313 is the smallest term such that 3313 +- 6 are both prime.

			71 is a term because it is the initial member of 5 consecutive primes {71, 73, 79, 83, 89} and (71 + 73 + 79 + 83 + 89)/5 = 79.
271 is a term because it is the initial member of 5 consecutive primes {271, 277, 281, 283, 293} and (271 + 277 + 281 + 283 + 293)/5 = 281.

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

Cf. A006562, A034964, A122535.

A293395:= proc(n)local a, b, c, d, e; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); d:=ithprime(n+3); e:=ithprime(n+4); if (a + b + d + e)/4 = c then RETURN (a); fi; end: seq(A293395(n), n=1..3000);

Select[Prime@ Range[1200], #[[3]] == Mean@ Delete[#, 3] &@ NestList[NextPrime, #, 4] &] (* Michael De Vlieger, Oct 09 2017 *)
Select[Partition[Prime[Range[1200]],5,1],Mean[#]==#[[3]]&][[;;,1]] (* Harvey P. Dale, Jul 31 2025 *)

for(n=1, 1000, a=prime(n); b=prime(n+1); c=prime(n+2); d=prime(n+3); e=prime(n+4); if((a+b+d+e)/4==c, print1(a,", ")));

list(lim)=my(v=List(),p=2,q=3,r=5,s=7); forprime(t=11,lim, if(p+q+s+t==4*r, listput(v,p)); p=q; q=r; r=s; s=t); Vec(v) \\ Charles R Greathouse IV, Oct 09 2017

Definiyion simplified by David A. Corneth, Oct 14 2017

Examples clarified by Harvey P. Dale, Jul 31 2025

A300365 Balanced primes of order fourteen.

Original entry on oeis.org

5297, 15647, 22073, 22501, 26309, 34721, 43499, 44111, 48809, 57529, 58171, 66797, 69151, 70199, 74551, 76493, 86959, 91297, 93169, 93199, 94343, 102217, 110777, 112289, 113093, 132361, 133493, 135461, 139921, 146021, 155303, 156521, 162557, 163753, 163789

Offset: 1

Muniru A Asiru, Mar 04 2018

nonn

			5297 is a member because 5297 = 5167 + 5171 + 5179 + 5189 + 5197 + 5209 + 5227 + 5231 + 5233 + 5237 + 5261 + 5273 + 5279 + 5281 + 5297 + 5303 + 5309 + 5323 + 5333 + 5347 + 5351 + 5381 + 5387 + 5393 + 5399 + 5407 + 5413 + 5417 + 5419  = 153613/29.

Muniru A Asiru, Table of n, a(n) for n = 1..6600

Cf. Balanced primes of order b: A006562 (b=1), A082077 (b=2), A082078 (b=3), A082079 (b=4), A096697 (b=5), A096698 (b=6), A096699 (b=7), A096700 (b=8), A096701 (b=9), A096702 (b=10), A096703 (b=11), A096704 (b=12), A300364 (b=13) this sequence (b=14).

P:=Filtered([1..200000],IsPrime);;
a:=List(Filtered(List([0..17000],k->List([1..29],j->P[j+k])),i->Sum(i)/29=i[15]),m->m[15]);

Module[{bal=14,nn=16000},Select[Partition[Prime[Range[nn]],2bal+1,1],Mean[#]==#[[bal+1]]&]][[;;,15]] (* Harvey P. Dale, Jul 07 2023 *)

A303093 Balanced primes of order one ending in 3.

Original entry on oeis.org

53, 173, 263, 373, 563, 593, 653, 733, 1103, 1123, 1223, 1753, 2903, 2963, 3313, 3733, 4013, 4993, 5113, 5303, 5393, 5563, 6073, 6263, 6323, 6373, 6863, 7523, 7583, 7823, 8713, 9473, 10253, 10853, 11903, 11933, 12583, 12653, 12973, 13043, 13463, 14543, 14753

Offset: 1

Muniru A Asiru, Apr 18 2018

			53 = (47 + 53 + 59)/3 = 159/3 and 53 = 5*10 + 3.

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

Intersection of A006562 and A030431.

Cf. A303092, A303094, A303095.

P:=Filtered([1..15000],IsPrime);;
a:=Filtered(List(Filtered(List([0..Length(P)-3],k->List([1..3],j->P[j+k])),i->Sum(i)/3=i[2]),m->m[2]),l-> l mod 10=3);

p:=ithprime: a:=n->`if`(add(p(n-k),k=-1..1)=3*p(n) and modp(p(n), 10) = 3,p(n),NULL): seq(a(n),n=3..2000);

Select[Partition[Prime[Range[2000]],3,1],Mean[#]==#[[2]]&&Mod[#[[2]],10]==3&][[All, 2]] (* Harvey P. Dale, Apr 09 2022 *)

A054801 Second term of balanced prime quartets: p(m)-p(m-1) = p(m+1)-p(m) = p(m+2)-p(m+1).

Original entry on oeis.org

257, 1747, 3307, 5107, 5387, 6317, 6367, 12647, 13457, 14747, 15797, 15907, 17477, 18217, 19477, 23327, 26177, 30097, 30637, 53617, 56087, 62207, 63697, 71347, 74471, 75527, 76561, 77557, 78797, 80917, 82787, 83437, 84437, 89107, 89387

Offset: 1

Henry Bottomley, Apr 10 2000

nonn

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

Cf. A006562, A054800-A054840.

Transpose[Select[Partition[Prime[Range[9000]],4,1],Length[ Union[ Differences[#]]] == 1&]][[2]] (* Harvey P. Dale, Oct 22 2013 *)

cp's OEIS Frontend

A053711 Numbers d such that, for some k, the upper and lower primes closest to k! are k! + d and k! - d.

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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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A085909 Smallest prime p>prime(n) such that p+prime(n+1)-prime(n) is the next prime after p; or a(n)=0 if no such prime exists.

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A126557 Primes in A126556.

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A178954 Primes prime(j) which cannot be written as 2*prime(j) = prime(j+k) + prime(j-k) for any 0 < k < j.

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A184247 Primes, q, such that for three consecutive primes, p, q & r, with p

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A293395 The initial member of 5 consecutive primes whose arithmetic mean is the middle member.

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Maple

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A300365 Balanced primes of order fourteen.

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