cp's OEIS Frontend

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)].

%I A073615 #16 Oct 19 2017 10:38:26
%S A073615 5,11,13,101,223,1277,1279,1616603,3772637849,51448351,
%T A073615 115438255651987,1007002660975121,433241801791933
%N 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)].
%e A073615 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.
%e A073615 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.
%e A073615 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.
%e A073615 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.
%e A073615 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.
%t A073615 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}]
%o A073615 (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
%Y A073615 See A236411 for a very similar sequence.
%K A073615 hard,more,nonn
%O A073615 2,1
%A A073615 _Amarnath Murthy_, Aug 06 2002
%E A073615 Edited and extended by _Robert G. Wilson v_, Aug 09 2002
%E A073615 Edited by _N. J. A. Sloane_, Feb 05 2014 following suggestions from _Don Reble_, _T. D. Noe_ and _Harvey P. Dale_.
%E A073615 a(10) from _Ray Chandler_, Feb 05 2014
%E A073615 a(12)-a(14) from _Don Reble_, Feb 06 2014