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

1,1

Comments

a(43) = 147618899630. - Donovan Johnson

Links

Table of n, a(n) for n=1..37.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A001223, A080378, A080381.

Sequence in context: A065270 A065282 A274930 * A257513 A276849 A367210

Adjacent sequences: A080376 A080377 A080378 * A080380 A080381 A080382

Keyword

nonn

Author

Labos Elemer, Mar 04 2003

Extensions

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

Status

approved