cp's OEIS Frontend

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.

%I A256620 #39 May 22 2025 10:21:42
%S A256620 12,30,42,312,600,858,1032,1290,1698,2112,2688,3768,4218,4230,4260,
%T A256620 5850,6132,6552,6702,7212,7308,8292,9420,9930,11970,12042,12378,15972,
%U A256620 17190,17598,17922,19470,19890,21600,24180,26862,30012,30852,32118
%N 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.
%C A256620 This sequence is a subsequence A014574 (average of twin prime pairs).
%C A256620 All terms are multiples of 6. - _Zak Seidov_, Apr 25 2015
%H A256620 Karl V. Keller, Jr., <a href="/A256620/b256620.txt">Table of n, a(n) for n = 1..500000</a>
%H A256620 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>
%e A256620 For n=12: 5,7,11,13,17,19 are six consecutive primes with 13 = 11 + 2 and (5+7+17+19)/4=12.
%e A256620 For n=1032: 1019,1021,1031,1033,1039,1049 are six consecutive primes with 1033 = 1031 + 2 and (1019+1021+1039+1049)/4=1032.
%t A256620 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 *)
%o A256620 (Python)
%o A256620 from sympy import isprime,prevprime,nextprime
%o A256620 for i in range(5,200001,2):
%o A256620   if isprime(i) and isprime(i+2):
%o A256620     a = prevprime(i)
%o A256620     b = prevprime(a)
%o A256620     if a+b+nextprime(i,2)+nextprime(i,3) == 4*(i+1): print(i+1,end=', ')
%o A256620   else: continue
%Y A256620 Cf. A077800 (twin primes), A014574.
%K A256620 nonn
%O A256620 1,1
%A A256620 _Karl V. Keller, Jr._, Apr 24 2015
%E A256620 Typo in Name fixed by _Zak Seidov_, Apr 25 2015