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.

5, 2, 9, 15, 39, 32, 305, 51, 2631, 3685, 170, 1156, 8775, 98, 5295, 41914, 106469, 167115, 186917, 1098776, 187784, 976193, 1166047, 423098, 77442332, 2643158, 11004239, 36330320, 259652255, 307899596, 2573725031, 411764049, 4080634008, 14841740642, 6022532018, 17035372732, 35045523209

Offset: 1

Author

Labos Elemer, Mar 04 2003

Keywords

• nonn

Comments

a(43) = 147618899630. - Donovan Johnson

Examples

			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.
		

Crossrefs

Cf. A001223, A080378, A080381.

Programs

Mathematica

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}]
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 *)

Formula

a(n)=Min{x; Union[{Mod[A001223(x), 4], ..., Mod[A001223(x+n-1), 4]}]=2}

Extensions

a(20)-a(37) from Donovan Johnson, Nov 16 2010