A167918 a(n) is smallest index k > n of k-th prime with f(n,k):=(p(k)+p(k+1))/(p(n)+p(n+1)) an integer >=2 (n=1,2,...).

6, 5, 5, 7, 17, 10, 20, 13, 55, 17, 26, 44, 81, 41, 35, 102, 30, 43, 33, 34, 49, 66, 173, 42, 45, 127, 65, 66, 228, 52, 117, 253, 80, 61, 62, 89, 162, 94, 123, 177, 256, 212, 162, 137, 138, 112, 212, 122, 189, 89, 160, 162, 201, 170, 137, 99, 140, 142, 405, 146, 190, 109

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 15 2009

Keywords

• nonn

Comments

(1) It is conjectured that sequence is infinite.

(2) It is conjectured that f(n,k)=2 for infinite many cases.

(3) Note the new link between two consecutive primes and twin primes.

(4) Note many possible generalizations with other fraction types (p(k) + ... + p(k+s))/(p(n) + ... + p(n+t)).

(5) Open problems: (a) is f(n,k) bounded, (b) which integer values for f(n,k) are "possible".

Examples

			f(1,6) = (p(6) + p(7))/(p(1) + p(2)) = (13 + 17)/(2 + 3) = 6 gives a(1)=6;
f(18,162) = (p(162) + p(163))/(p(18) + p(19)) = (953 + 967)/(61 + 67) = 15 gives a(18)=162.
		

References

• Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer, 2005
• Harold Davenport, Multiplicative Number Theory, Springer-Verlag, New York, 1980
• Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications, 2005

Crossrefs

Cf. A000040 (the prime numbers).

Cf. A167790.

Programs

Maple

A001043 := proc(n) option remember; ithprime(n)+ithprime(n+1) ; end proc: A167918 := proc(n) local k ; for k from n+1 do if A001043(k) mod A001043(n) = 0 then return k; end if ; end do; end proc: seq(A167918(n),n=1..100) ; # R. J. Mathar, Nov 17 2009

Extensions

a(2), a(4), a(18) and a(20) corrected by R. J. Mathar, Nov 17 2009