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A259025 Numbers k such that k is the average of four consecutive primes k-11, k-1, k+1 and k+11.

%I A259025 #46 Feb 16 2025 08:33:25
%S A259025 420,1050,2028,2730,3582,4230,4242,4272,4338,6090,6132,6690,6792,8220,
%T A259025 11058,11160,11970,12252,15288,19542,19698,21588,21600,26892,27540,
%U A259025 28098,28308,29400,30840,30870,31080,32412,42072,45318,47808,48120
%N A259025 Numbers k such that k is the average of four consecutive primes k-11, k-1, k+1 and k+11.
%C A259025 This sequence is a subsequence of A014574 (average of twin prime pairs) and A256753.
%C A259025 The terms ending in 0 are congruent to 0 mod 30.
%C A259025 The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
%H A259025 Karl V. Keller, Jr., <a href="/A259025/b259025.txt">Table of n, a(n) for n = 1..10000</a>
%H A259025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%F A259025 a(n) = A052376(n) + 11. - _Robert G. Wilson v_, Jul 15 2015
%e A259025 For n=420: 409, 419, 421, 431 are consecutive primes (n-11=409, n-1=419, n+1=421, n+11=431).
%e A259025 For n=1050: 1039, 1049, 1051, 1061 are consecutive primes (n-11=1039, n-1=1049, n+1=1051, n+11=1061).
%t A259025 {p, q, r, s} = {2, 3, 5, 7}; lst = {}; While[p < 50000, If[ Differences[{p, q, r, s}] == {10, 2, 10}, AppendTo[lst, q + 1]]; {p, q, r, s} = {q, r, s, NextPrime@ s}]; lst (* _Robert G. Wilson v_, Jul 15 2015 *)
%t A259025 Mean/@Select[Partition[Prime[Range[5000]],4,1],Differences[#]=={10,2,10}&] (* _Harvey P. Dale_, Sep 11 2019 *)
%o A259025 (Python)
%o A259025 from sympy import isprime,prevprime,nextprime
%o A259025 for i in range(0,50001,2):
%o A259025   if isprime(i-1) and isprime(i+1):
%o A259025     if prevprime(i-1) == i-11 and nextprime(i+1) == i+11 :  print (i,end=', ')
%o A259025 (PARI) is(n)=n%6==0&&isprime(n-11)&&isprime(n-1)&&isprime(n+1)&&isprime(n+11)&&!isprime(n-7)&&!isprime(n-5)&&!isprime(n+5)&&!isprime(n+7) \\ _Charles R Greathouse IV_, Jul 17 2015
%Y A259025 Cf. A052376, A077800 (twin primes), A014574, A249674 (30n), A256753.
%K A259025 nonn
%O A259025 1,1
%A A259025 _Karl V. Keller, Jr._, Jun 16 2015