cp's OEIS Frontend

A272856 Greatest length of a chain of consecutive primes p==1 (mod 3) for which A261029 (2*prime(n)*p) is 4-i if prime(n) == i (mod 3), where i=1,2.

%I A272856 #13 Sep 05 2018 02:28:58
%S A272856 5,7,16,19,32,35,50,76,81,108,140,139,171,206,254,259,305,346,349,404,
%T A272856 449,504,582,634,645,699,707,772,930,1006,1078,1097,1258,1271,1362,
%U A272856 1448,1529,1633,1737,1752,1951,1970,2064,2082,2310,2550,2659,2672,2783,2917
%N A272856 Greatest length of a chain of consecutive primes p==1 (mod 3) for which A261029 (2*prime(n)*p) is 4-i if prime(n) == i (mod 3), where i=1,2.
%e A272856 Let n=3; then prime(n)=5. Since 5 == 2 (mod 3), i=2. So a(3) is the greatest length of a chain of consecutive primes p == 1 (mod 3) for which A261029(10*p) = 4 - 2 = 2. So these primes are in A272381. The first term is 7, and we have the chain of consecutive primes == 1 (mod 3): {7, 13, 19, 31, 37}. Since the following prime 43 == 1 (mod 3) is not in A272381, the chain ends and its length is 5. The second chain is the singleton {71}. So a(3)=5.
%t A272856 a261029[n_]:=a261029[n]={x,y,z}/.{ToRules[Reduce[x^3+y^3+z^3-3 x y z==n&&0<=x<=y<=z&&z>=x+1,Integers]]}/.{x,y,z}->{};
%t A272856 data={};
%t A272856 Do[p=Prime[n];
%t A272856 primes=Select[Prime[Range[1+PrimePi[(2p)^2]]],Mod[#,3]==1&];
%t A272856 tmp=Map[{#,Length[a261029[2 # p]]}&,primes];
%t A272856 AppendTo[data,{{n,2p,1+Mod[2p,3]},{{Length[#],Max[Map[Length,Select[Split[Differences[Flatten[Map[Position[primes,#,1,1]&,#]]]],#[[1]]==1&]]+1]},#}&[Map[#[[1]]&,Select[tmp,#[[2]]==(1+Mod[2p,3])&]]]}];Print[Last[data]],{n,3,10}]
%t A272856 Map[Length[a261029[#]]&,Range[0,20]] (* A261029 *)
%t A272856 Last[Last[data[[1]]]] (* A272381 *)
%t A272856 Last[Last[data[[2]]]] (* A272382 *)
%t A272856 Last[Last[data[[3]]]] (* A272384 *)
%t A272856 Last[Last[data[[4]]]] (* A272404 *)
%t A272856 Last[Last[data[[5]]]] (* A272406 *)
%t A272856 Last[Last[data[[6]]]] (* A272407 *)
%t A272856 Last[Last[data[[7]]]] (* A272409 *)
%t A272856 Map[#[[2]][[1]][[1]]&,data] (* A268665 *)
%t A272856 Map[#[[2]][[1]][[2]]&,data] (* A272856 *)
%Y A272856 Cf. A261029, A268665, A272381, A272382, A272384, A272404, A272406, A272407, A272409.
%K A272856 nonn
%O A272856 3,1
%A A272856 _Vladimir Shevelev_ and _Peter J. C. Moses_, May 08 2016