A068843 Smallest prime in the first occurrence of a nondecreasing difference for a set of exactly n successive primes.

37, 11, 17, 2, 1091, 2897, 13451, 448363, 7407287, 34400141, 255030533, 6564959561, 45605475961, 121054164221, 2552790756469

Offset

1,1

Comments

This is the same as asking for the smallest prime in the first occurrence of a nonnegative second difference for a set of n successive primes.

Links

Table of n, a(n) for n=1..15.

Zak Seidov, Consecutive non-decreasing prime gaps, SeqFan list, May 16 2017

Example

The prime 2 starts the first run of exactly 4 nondecreasing gaps (1, 2, 2, 4) between the 5 primes (2, 3, 5, 7, 11). (The next gap would be of 2, smaller than 4.) Therefore a(4) = 2.
The prime 11 starts the first run of exactly 2 nondecreasing gaps (2, 4) between the 3 primes (11, 13, 17). (The preceding gap is 4 > 2 and the next gap would be 2 < 4.) Therefore a(2) = 11. The sequences of primes (2, 3, 5) as well as (5, 7, 11) are part of a longer run of nondecreasing gaps and are therefore not considered for the case n = 2.
The prime 17 starts the first run of exactly 3 nondecreasing gaps (2, 4, 4) between the 4 primes (17, 19, 23, 29). (The preceding gap is 4 > 2 and the next gap would be 2 < 4.) Therefore a(3) = 17. Again, primes which are part of a longer run cannot be considered.
The prime 37 starts the first run of exactly 1 nondecreasing gap (4) between the primes 37 and 41. (The preceding gap is 6 > 4 and the next gap would be 2 < 4.) Therefore a(1) = 37.

Mathematica

(* used to find a(11) *) k = 0; While[p = Select[ Range[ k*10^6, (k + 1)*10^6 + 10^4], PrimeQ[ # ] & ]; l = Length[p]; d1 = Take[p, 1 - l] - Take[p, l - 1]; d2 = Take[d1, 2 - l] - Take[d1, l - 2]; s = Sign[ Sign[d2] + 1]; q = StringPosition[ ToString[s], StringDrop[ StringDrop[ ToString[ Table[1, {10}]], 1], -1]]; q == {}, k++ ]; p[[ (q[[1, 1]] + 1)/3 ]]

Prog

(PARI) A068843(n, c=n+1, g, o=2, P=2)={forprime(p=3, , c--; g>(g=-o+o=p)||next; c||break; c=n; P=p-g); P} \\ M. F. Hasler, May 16 2017

Crossrefs

Cf. A158939, A158940.

Sequence in context: A107812 A254324 A356745 * A284497 A033357 A093860

Adjacent sequences: A068840 A068841 A068842 * A068844 A068845 A068846

Keyword

nonn,base,hard,more

Author

Amarnath Murthy, Mar 10 2002

Extensions

More terms from Robert G. Wilson v, May 10 2002 and Dec 08 2002, who finds that a(12) > 2.7*10^9

a(12) = 6564959561 from Zak Seidov and other edits by M. F. Hasler, May 16 2017

a(13)-a(15) from Giovanni Resta, May 18 2017

Status

approved