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A084299 Smallest primes such that the subsequent terms of consecutive prime differences (A001223) modulo 6 (A054763) displays repeatedly n times a {0,2,4} pattern of remainders of differences.

%I A084299 #11 Nov 05 2024 03:14:45
%S A084299 83,2903,5897,319499,346943,7974179,15262433,33954251,5521833683,
%T A084299 83993232497,848099080883,1293322433639
%N A084299 Smallest primes such that the subsequent terms of consecutive prime differences (A001223) modulo 6 (A054763) displays repeatedly n times a {0,2,4} pattern of remainders of differences.
%e A084299 For n=1: a(1) = 83 is followed by [6,8,4].
%e A084299 For n=2: a(2) = 2903 is followed by [6,2,4,18,2,4].
%e A084299 For n=3: a(3) = 5897 is followed by [6,20,4,12,14,28,6,20,4].
%e A084299 For n=4: a(4) = 319499 is followed by [12,8,22,6,20,10,12,2,10,6,32,34].
%e A084299 For n=5: a(5) = 346943 is followed by [18,2,40,....,30,2,10] differences corresponding to n "wavelet" of [0,2,4] remainders modulo 6.
%t A084299 (* generates a(5) *) d[x_] := Prime[x+1]-Prime[x]; md[x_] := Mod[Prime[x+1]-Prime[x], 6]; h={k1=0, k2=2, k3=4}; k=0; Do[If[Equal[md[n], k1]&&Equal[md[n+1], k2]&& Equal[md[n+2], k3]&&Equal[md[n+3], k1]&&Equal[md[n+4], k2]&&Equal[md[n+5], k3] &&Equal[md[n+6], k1]&&Equal[md[n+7], k2]&&Equal[md[n+8], k3] &&Equal[md[n+9], k1]&&Equal[md[n+10], k2]&&Equal[md[n+11], k3]&& Equal[md[n+12], k1]&&Equal[md[n+13], k2]&&Equal[md[n+14], k3], k=k+1; Print[{k, n, Prime[n], Table[md[n+j], {j, -1, 15}], Table[d[n+j], {j, -1, 15}]}]], {n, 2, 10000000}]
%o A084299 (PARI) lista(pmax) = {my(rec = 0, m = 0, c = 0, prv = 2, p0 = 0, d); forprime(p = 3, pmax, d = (p-prv)%6; if(d == 0 && m == 0, p0 = prv); if(d == c, m++; c = (c+2)%6; if(!(m%3) && m/3 > rec, print1(p0, ", "); rec++; m = 0), if(d == 0, p0 = prv; c = 2; m = 1, c = 0; m = 0)); prv = p);} \\ _Amiram Eldar_, Nov 04 2024
%Y A084299 Cf. A001223, A054763, A016045.
%K A084299 nonn,more
%O A084299 1,1
%A A084299 _Labos Elemer_, Jun 02 2003
%E A084299 a(9)-a(12) from _Amiram Eldar_, Nov 04 2024