A052378 -id:A052378 - cp's OEIS frontend

A258088 Numbers n such that n is the average of four consecutive primes n-5, n-1, n+1 and n+5.

12, 18, 42, 102, 108, 228, 312, 462, 858, 882, 1092, 1302, 1428, 1488, 1872, 1998, 2688, 3462, 4518, 4788, 5232, 5652, 6828, 7878, 8292, 10458, 13692, 13878, 15732, 16062, 16068, 16188, 17388, 19422, 19428, 20748, 21018, 21318, 22278, 23058

Offset: 1

Karl V. Keller, Jr., May 19 2015

nonn

Previous name was: Numbers n such that n is the average of some twin prime pair p, q (q=p+2) (i.e., n=p+1=q-1) where p-4, p, q, and q+4 are consecutive primes.

This is a subsequence of A014574 (average of twin prime pairs) and A256753.

			12 is the average of the four consecutive primes 7, 11, 13, 17.
18 is the average of the four consecutive primes 13, 17, 19, 23.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A052378, A077800 (twin primes), A256753.

a={};Do[If[Prime[x + 3] - Prime[x]==10, AppendTo[a, Prime[x]+ 5]], {x, 1, 4000}]; a (* Vincenzo Librandi, Jul 18 2015 *)
Mean/@Select[Partition[Prime[Range[3000]],4,1],Differences[#]=={4,2,4}&] (* Harvey P. Dale, Sep 18 2018 *)

is(n)=isprime(n-5)&&isprime(n-1)&&isprime(n+1)&&isprime(n+5) \\ Charles R Greathouse IV, Aug 28 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,50001,2):
  if isprime(i-1) and isprime(i+1):
    if prevprime(i-1) == i-5 and nextprime(i+1) == i+5: print (i,end=', ')

a(n) = A052378(n) + 5. - Karl V. Keller, Jr., Jul 17 2015

New name from Karl V. Keller, Jr., Jul 21 2015

A261541 Least positive integer m such that both m and m*n belong to the set {k>0: prime(k)+2, prime(k)+6, prime(k)+8 are all prime}.

Original entry on oeis.org

3, 358712, 34772, 79631, 1822685, 22865, 2066, 2593722, 26, 3418900, 26, 711611, 286, 1493190, 882854, 513312, 1707237, 788232, 913695, 1980985, 7147, 443152, 479580, 2589105, 865432, 265243, 103641, 160536, 398360, 851672

Offset: 1

Zhi-Wei Sun, Aug 24 2015

nonn

Conjecture: (i) Each positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2, prime(k)+6 and prime(k)+8 are all prime}.
(ii)  Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+4, prime(k)+6 and prime(k)+10 are all prime}.
For example, 3/4 = 20723892/27631856, and prime(20723892)+2 = 387875561+2 = 387875563,  prime(20723892)+6 = 387875567, prime(20723892)+8 = 387875569, prime(27631856)+2 = 525608591+2 =525608593, prime(27631856)+6 = 525608597,  prime(27631856)+8 = 525608599 are all prime.  Also, 3/4 = 599478/799304, and prime(599478)+4 = 8951857+4 = 8951861, prime(599478)+6 = 8951863, prime(599478)+10 = 8951867, prime(799304)+4 = 12183943+4 = 12183947, prime(799304)+6 = 12183949, prime(799304)+10 = 12183953 are all prime.
Part (i) of the conjecture implies that there are infinitely many primes p with p+2, p+6 and p+8 all prime, while part (ii) implies that there are infinitely many primes p with p+4, p+6 and p+10 all prime.

			a(1) = 3 since 3*1 = 3, and prime(3)+2 = 5+2 =7, prime(3)+6 = 11 and prime(3)+8 = 13 are all prime.
a(2) = 358712 since prime(358712)+2 = 5158031+2 = 5158033, prime(358712)+6 = 5158037, prime(358712)+8 = 5158039, prime(358712*2)+2 = 10852601+2 = 10852603, prime(358712*2)+6 = 10852607 and prime(358712*2)+8 = 10852609 are all prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Checking part (i) of the conjecture for r = a/b with a,b = 1..25
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A007530, A052378, A236511, A259487, A259540.

f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&PrimeQ[f[k]+6]&&PrimeQ[f[k]+8]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,30}]

A384299 Primes p such that p + 8, p + 12 and p + 20 are also primes.

Original entry on oeis.org

11, 59, 89, 389, 479, 1439, 1559, 1601, 2531, 2699, 3209, 3449, 3911, 5639, 5849, 7529, 8081, 8669, 10091, 12269, 12401, 12899, 13151, 14411, 14759, 17021, 19421, 21011, 21851, 22271, 23189, 25931, 26099, 28649, 28859, 31139, 31469, 33191, 33569, 36551, 39659, 40751, 42689, 43391, 43781, 44111

Offset: 1

Alexander Yutkin, May 25 2025

nonn

Initial members of prime quartets that correspond to the difference pattern [8, 4, 8].

			p=89: 89+8=97, 89+12=101, 89+20=109 —> prime quartet: (89, 97, 101, 109).

Cf. A000040, A001223.

Cf. A136162 [2, 4, 2], A052378 [4, 2, 4], A382810 [6, 4, 6].

q:= n-> andmap(i-> isprime(n+4*i), [0,2,3,5]):
select(q, [5+6*i$i=1..7351])[];  # Alois P. Heinz, May 29 2025

Select[Prime[Range[4591]],AllTrue[#+{8,12,20},PrimeQ]&] (* James C. McMahon, May 29 2025 *)

A383396 Primes p such that p + 6, p + 10, p + 12, p + 16 and p + 22 are also primes.

Original entry on oeis.org

7, 31, 2677, 35521, 42451, 44257, 55807, 93481, 118891, 198817, 221707, 234181, 313981, 393571, 560227, 669847, 1107781, 1210387, 1596367, 1616611, 1738411, 2710921, 3194551, 3377587, 3441931, 3484561, 3586537, 3699181, 3887551, 3904897, 4095661, 4192261, 4239721

Offset: 1

Alexander Yutkin, Apr 25 2025

nonn

Initial members of prime sextuples that correspond to the difference pattern [6, 4, 2, 4, 6].

			p = 2677: 2677 + 6 = 2683, 2677 + 10 = 2687, 2677 + 12 = 2689, 2677 + 16 = 2693, 2677 + 22 = 2699 -> prime sextuple: (2677, 2683, 2687, 2689, 2693, 2699).

Cf. A000040, A001223.

Cf. A052378 [4, 2, 4], A022008 [4, 2, 4, 2, 4].

Select[Prime[Range[298900]], AllTrue[#+{6,10,12,16,22}, PrimeQ]&] (* James C. McMahon, May 02 2025 *)

a(n) == 1 (mod 6).

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A258088 Numbers n such that n is the average of four consecutive primes n-5, n-1, n+1 and n+5.

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A261541 Least positive integer m such that both m and m*n belong to the set {k>0: prime(k)+2, prime(k)+6, prime(k)+8 are all prime}.

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A384299 Primes p such that p + 8, p + 12 and p + 20 are also primes.

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A383396 Primes p such that p + 6, p + 10, p + 12, p + 16 and p + 22 are also primes.

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