OFFSET
1,1
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".
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
EXAMPLE
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.
MAPLE
CROSSREFS
KEYWORD
nonn
AUTHOR
Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 15 2009
EXTENSIONS
a(2), a(4), a(18) and a(20) corrected by R. J. Mathar, Nov 17 2009
STATUS
approved