A256620 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 arithmetic mean of the four numbers consisting of the two primes before p and the two primes after q.

12, 30, 42, 312, 600, 858, 1032, 1290, 1698, 2112, 2688, 3768, 4218, 4230, 4260, 5850, 6132, 6552, 6702, 7212, 7308, 8292, 9420, 9930, 11970, 12042, 12378, 15972, 17190, 17598, 17922, 19470, 19890, 21600, 24180, 26862, 30012, 30852, 32118

Offset

1,1

Comments

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

All terms are multiples of 6. - Zak Seidov, Apr 25 2015

Links

Karl V. Keller, Jr., Table of n, a(n) for n = 1..500000

Eric Weisstein's World of Mathematics, Twin Primes

Example

For n=12: 5,7,11,13,17,19 are six consecutive primes with 13 = 11 + 2 and (5+7+17+19)/4=12.
For n=1032: 1019,1021,1031,1033,1039,1049 are six consecutive primes with 1033 = 1031 + 2 and (1019+1021+1039+1049)/4=1032.

Mathematica

avQ[lst_]:=Module[{td=TakeDrop[lst, {3, 4}]}, Mean[td[[1]]]==Mean[td[[2]]] && td[[1, 2]]-td[[1, 1]]==2]; Mean[Take[#, {3, 4}]]&/@Select[Partition[ Prime[ Range[ 3500]], 6, 1], avQ] (* The program uses the TakeDrop function from Mathematica version 10.2 *) (* Harvey P. Dale, Jul 16 2015 *)

Prog

(Python)
from sympy import isprime, prevprime, nextprime
for i in range(5, 200001, 2):
..if isprime(i) and isprime(i+2):
....a = prevprime(i)
....b = prevprime(a)
....if a+b+nextprime(i, 2)+nextprime(i, 3) == 4*(i+1): print(i+1, end=', ')
..else: continue

Crossrefs

Cf. A077800 (twin primes), A014574.

Sequence in context: A364975 A298077 A135502 * A125582 A031107 A019557

Adjacent sequences: A256617 A256618 A256619 * A256621 A256622 A256623

Keyword

nonn

Author

Karl V. Keller, Jr., Apr 24 2015

Extensions

Typo in Name fixed by Zak Seidov, Apr 25 2015

Status

approved