A031217 Number of terms in longest arithmetic progression of consecutive primes starting at n-th prime (conjectured to be unbounded).

2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2

Offset

1,1

Comments

a(n) <= 4 for n <= 10^5. - Reinhard Zumkeller, Feb 02 2007

The first instance of 4 consecutive primes in an arithmetic progression is (251, 257, 263, 269), which starts with the 54th prime. The first instance of 5 consecutive primes in an arithmetic progression is (9843019, 9843049, 9843079, 9843109, 9843139), which starts with the 654926th prime. [Harvey P. Dale, Jul 13 2011]

References

R. K. Guy, Unsolved Problems in Number Theory, A6.

Links

R. Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for sequences related to primes in arithmetic progressions

Example

At 47 there are 3 consecutive primes in A.P., 47 53 59.

Mathematica

max = 5; a[n_] := Catch[pp = NestList[ NextPrime, Prime[n], max-1]; Do[ If[ Length[ Union[ Differences[pp[[1 ;; -k]] ] ] ] == 1, Throw[max-k+1]], {k, 1, max-1}]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jul 17 2012 *)
Length[Split[Differences[#]][[1]]]&/@Partition[Prime[Range[120]], 10, 1]+1 (* Harvey P. Dale, Mar 17 2024 *)

Prog

(PARI) a(n)=my(p=prime(n), q=nextprime(p+1), g=q-p, k=2); while(nextprime(q+1)==q+g, q+=g; k++); k \\ Charles R Greathouse IV, Jun 20 2013

Crossrefs

Cf. A001223.

Sequence in context: A300666 A120881 A297930 * A064131 A368859 A338238

Adjacent sequences: A031214 A031215 A031216 * A031218 A031219 A031220

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers

Status

approved