A167918 - OEIS

%I #5 May 11 2019 00:12:04

%S 6,5,5,7,17,10,20,13,55,17,26,44,81,41,35,102,30,43,33,34,49,66,173,

%T 42,45,127,65,66,228,52,117,253,80,61,62,89,162,94,123,177,256,212,

%U 162,137,138,112,212,122,189,89,160,162,201,170,137,99,140,142,405,146,190,109

%N 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,...).

%C (1) It is conjectured that sequence is infinite.

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

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

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

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

%D Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer, 2005

%D Harold Davenport, Multiplicative Number Theory, Springer-Verlag, New York, 1980

%D Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications, 2005

%e f(1,6) = (p(6) + p(7))/(p(1) + p(2)) = (13 + 17)/(2 + 3) = 6 gives a(1)=6;

%e f(18,162) = (p(162) + p(163))/(p(18) + p(19)) = (953 + 967)/(61 + 67) = 15 gives a(18)=162.

%p 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

%Y Cf. A000040 (the prime numbers).

%Y Cf. A167790.

%K nonn

%O 1,1

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

%E a(2), a(4), a(18) and a(20) corrected by _R. J. Mathar_, Nov 17 2009