A073615 Let p(k) denote the k-th prime; a(n) = smallest p(m) > p(n) such that the n-1 differences between [p(n), p(n+1), ..., p(2n-1)] are the same as the n-1 differences between [p(m), p(m+1), ..., p(m+n-1)].

5, 11, 13, 101, 223, 1277, 1279, 1616603, 3772637849, 51448351, 115438255651987, 1007002660975121, 433241801791933

Offset

2,1

Links

Table of n, a(n) for n=2..14.

Example

Consider the prime sequence starting from 2: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ... The first difference sequence is 1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, ... There exists no prime sequence > 2 which has a first difference of 1.
Therefore we set offset = 2 and begin with the second prime (3, with its first forward difference of 2) and look for the first prime above 3 with a forward difference of 2. That number is 5, so a(2) = 5.
Next, n=3, start now with the third prime p(3)=5, with two forward differences of 2,4. The next prime above 5 which starts out with differences of 2,4 is 11. So a(3) = 11.
Next start with the fourth prime - 7, with three forward differences of 4,2,4. The next prime above 7 which starts out with those difference is 13. So a(4) = 13.
a(5) = 101: The differences between the primes 101, 103, 107, 109 & 113 are 2, 4, 2 & 4, the first to match the differences between 11, 13, 17, 19 & 23.

Mathematica

Do[s = Table[Prime[i + 1] - Prime[i], {i, n + 1, 2n}]; p = 0; q = 0; a = s; k = n + 2; While[p = q; q = Prime[k]; a = Drop[a, 1]; a = Append[a, q - p]; s != a, k++ ]; Print[Prime[PrimePi[q] - n]], {n, 1, 8}]

Prog

(PARI) A073615 = n->{d=vector(n-1, i, prime(n+i)-prime(n)); forprime(pm=prime(n+1), 9e9, for(k=1, #d, isprime(pm+d[k])||next(2)); p=pm; for(k=1, #d, (pm+d[k]==p=nextprime(p+1))||next(2)); return(pm))} \\ Yields a(8) in 0.1 sec, but is too slow beyond that. - M. F. Hasler, Feb 05 2014

Crossrefs

See A236411 for a very similar sequence.

Sequence in context: A348313 A352534 A236411 * A346147 A275118 A275640

Adjacent sequences: A073612 A073613 A073614 * A073616 A073617 A073618

Keyword

hard,more,nonn

Author

Amarnath Murthy, Aug 06 2002

Extensions

Edited and extended by Robert G. Wilson v, Aug 09 2002

Edited by N. J. A. Sloane, Feb 05 2014 following suggestions from Don Reble, T. D. Noe and Harvey P. Dale.

a(10) from Ray Chandler, Feb 05 2014

a(12)-a(14) from Don Reble, Feb 06 2014

Status

approved