A167790 -id:A167790 - cp's OEIS frontend

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

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

nonn

(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".

			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.

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

Cf. A000040 (the prime numbers).

Cf. A167790.

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

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

A168069 a(n) is the index k of the smallest k-th prime p(k) with f(k,n):=(p(k) + p(k+1))/n an integer (n = 1,2,3,...)

Original entry on oeis.org

1, 2, 3, 2, 1, 3, 8, 2, 4, 6, 13, 3, 9, 8, 6, 16, 11, 4, 21, 10, 8, 13, 19, 5, 14, 9, 22, 16, 69, 6, 24, 18, 25, 11, 7, 29, 21, 12, 17

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 18 2009

nonn

(1) EVERY natural k appears in the sequence, some more than once
(2) Theoretical interest for cases (I) a(n) < n, (II) a(n) = n, (I) a(n) > n
(3) Note cases n=1, 3, 16, ... with a(n) = n

			(1) p(1)+p(2)=2+3=1 x 5 gives a(1)=a(5)=1
(2) p(3)+p(4)=5+7=2^2 x 3 gives a(3)=a(6)=a(12)=3, but a(2)=2 < 3, because p(2)+p(2)=2 x 2^2
(3) p(16)+p(17)=53+59=2^4 x 7=16 x 7 gives a(16)=16
(4) p(69)+p(70)=347+349=2^3 x 3 x 29 gives a(29)=69

Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005

A000040 The prime numbers

A167790

cp's OEIS Frontend

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,...).

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