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A080379 Least n such that n consecutive values in A080378 equals 2; i.e., exactly n differences between consecutive primes give residues 2 when divided by 4.

%I A080379 #15 Dec 01 2022 15:47:22
%S A080379 5,2,9,15,39,32,305,51,2631,3685,170,1156,8775,98,5295,41914,106469,
%T A080379 167115,186917,1098776,187784,976193,1166047,423098,77442332,2643158,
%U A080379 11004239,36330320,259652255,307899596,2573725031,411764049,4080634008,14841740642,6022532018,17035372732,35045523209
%N A080379 Least n such that n consecutive values in A080378 equals 2; i.e., exactly n differences between consecutive primes give residues 2 when divided by 4.
%C A080379 a(43) = 147618899630. - _Donovan Johnson_
%F A080379 a(n)=Min{x; Union[{Mod[A001223(x), 4], ..., Mod[A001223(x+n-1), 4]}]=2}
%e A080379 n=4: a(4)=15,differences between {47,53,59,61,67} are {6,6,2,6} corresponds to exactly four differences congruent to 2 mod 4,since before and after 47-43=4 or 71-67=4 are congruent to 0 mod 4.
%t A080379 dp[x_] := Mod[Prime[x+1]-Prime[x], 4] pat[x_, h_] := Table[dp[x+j], {j, 0, h-1}] up[x_, h_] := Union[pat[x, h]] Table[fa=1; k=0; Do[s=up[n, h]; s1=Length[s]; s2=Part[u=pat[n+1, h], Length[u]]; s3=Part[w=pat[n-1, h], 1]; If[Equal[s1, 1]&&Equal[fa, 1]&&Equal[s2, 0]&&Equal[s3, 0], k=k+1; Print[{k, h, n, Prime[n], s, s1}]; fa=0], {n, 2, 200000}], {h, 1, 19}]
%t A080379 With[{c=Mod[Differences[Prime[Range[12*10^5]]],4]},Join[{5,2},Drop[ Flatten[ Table[ SequencePosition[ c,Join[ {0},PadRight[ {},n,2],{0}],1][[All,1]],{n,0,25}]]+1,3]]] (* The program generates the first 24 terms of the sequence. *) (* _Harvey P. Dale_, Dec 01 2022 *)
%Y A080379 Cf. A001223, A080378, A080381.
%K A080379 nonn
%O A080379 1,1
%A A080379 _Labos Elemer_, Mar 04 2003
%E A080379 a(20)-a(37) from _Donovan Johnson_, Nov 16 2010