A256753 Numbers n such that n is both the average of some twin prime pair p, q (q = p+2) (i.e., n = p+1 = q-1) and is also the average of the prime before p and the prime after q.

12, 18, 30, 42, 60, 102, 108, 228, 270, 312, 420, 462, 570, 600, 858, 882, 1050, 1092, 1230, 1290, 1302, 1428, 1488, 1620, 1872, 1998, 2028, 2340, 2550, 2688, 2730, 3390, 3462, 3540, 3582, 4020, 4230, 4242, 4272, 4338, 4518, 4650, 4788

Offset: 1

Author

Karl V. Keller, Jr., Apr 09 2015

Keywords

• nonn

Comments

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

Examples

			For n=12: 7, 11, 13, 17 are four consecutive primes with 13 = 11 + 2 and (7+17)/2 = 12.
For n=18: 13, 17, 19, 23 are four consecutive primes with 19 = 17 + 2 and (13+23)/2 = 18.
		

Links

• Karl V. Keller, Jr., Table of n, a(n) for n = 1..500000
• Eric Weisstein's World of Mathematics, Twin Primes

Crossrefs

Cf. A077800 (twin primes), A014574.

Programs

Mathematica

Select[Prime[Range[10^3]],PrimeQ[#+2]&&2*#+2==NextPrime[#,-1]+NextPrime[#,2]&]+1 (* Ivan N. Ianakiev, Apr 23 2015 *)
Select[Partition[Prime[Range[700]],4,1],#[[3]]-#[[2]]==2&&(#[[1]]+#[[4]])/2 == (#[[2]]+#[[3]])/2&][[All,2]]+1 (* Harvey P. Dale, May 06 2022 *)

PARI

lista(nn) = {forprime(p=3, nn, if (isprime(p+2), if (precprime(p-1)+nextprime(p+3) == 2*(p+1), print1(p+1, ", "));););} \\ Michel Marcus, Apr 12 2015

Python

from sympy import isprime,prevprime,nextprime
for i in range(5,12001,2):
  if isprime(i) and isprime(i+2):
    if prevprime(i)+nextprime(i,2) == 2*(i+1): print(i+1,end=', ')