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%I #23 Mar 17 2024 15:30:03
%S 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,
%T 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,
%U 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
%N Number of terms in longest arithmetic progression of consecutive primes starting at n-th prime (conjectured to be unbounded).
%C a(n) <= 4 for n <= 10^5. - _Reinhard Zumkeller_, Feb 02 2007
%C 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]
%D R. K. Guy, Unsolved Problems in Number Theory, A6.
%H R. Zumkeller, <a href="/A031217/b031217.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%e At 47 there are 3 consecutive primes in A.P., 47 53 59.
%t 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 *)
%t Length[Split[Differences[#]][[1]]]&/@Partition[Prime[Range[120]],10,1]+1 (* _Harvey P. Dale_, Mar 17 2024 *)
%o (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
%Y Cf. A001223.
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_
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