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

Author

Karl V. Keller, Jr., Apr 24 2015

Keywords

• nonn

Comments

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

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

Examples

			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.
		

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

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 *)

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

Extensions

Typo in Name fixed by Zak Seidov, Apr 25 2015