A187762 a(n) = smallest final prime in a chain of 2n+1 consecutive primes such that sum of the last 1, 3, 5, ..., 2n+1 terms in the sequence is also a prime.

2, 11, 17, 47, 71, 157, 157, 167, 203569, 203569, 2803083484951

Offset

0,1

Comments

Conjecture: a(n) exists for all n. (This may follow from the Green-Tao theorem, Dirichlet's theorem and Dickson's conjecture.)

Links

Table of n, a(n) for n=0..10.

Example

Consider the chain of following consecutive prime numbers 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157
Take the sum of an odd number of primes out of this sequence starting at the end:
S(1) = 157
S(3) = 157 + 151 + 149 = 457
S(5) = 157 + 151 + 149 + 139 + 137 = 733
S(7) = 157 + 151 + 149 + 139 + 137 + 131 +127 = 991
S(9) = 157 + 151 + 149 + 139 + 137 + 131 +127 + 113 + 109 = 1213
S(11) = 157 + 151 + 149 + 139 + 137 + 131 +127 + 113 + 109 + 107 + 103 = 1423
All of these are prime numbers.
Currently a(10) is the last known term, a chain of 21 primes found after searching up to 4*10^13. The 21 consecutive primes are 2803083484321, 2803083484343, 2803083484349, 2803083484363, 2803083484391, 2803083484429, 2803083484499, 2803083484507, 2803083484633, 2803083484637, 2803083484639, 2803083484673, 2803083484697, 2803083484703, 2803083484763, 2803083484777, 2803083484781, 2803083484819, 2803083484921, 2803083484937, 2803083484951, where the sums S(21), S(19), S(17), S(15) . . . . to S(1): 58864753177133, 53258586208469, 47652419239757, 42046252270937, 36440085301931, 30833918332661, 25227751363349, 19621584393949, 14015417424409, 8409250454809, 2803083484951 respectively are also primes.

Mathematica

(* This program is not convenient for n > 9 *) run[m_, n_] := Prime /@ Range[m + 2n, m, -1]; ok[ru_List] := (test = True; For[k = 1, k <= Length[ru], k = k+2, s = Total[ru[[1 ;; k]]]; If[! PrimeQ[s], test = False; Break[]]]; test); a[n_] := a[n] = Catch[For[m = 1, m <= 10^5, m++, r = run[m, n]; If[ok[r ], Throw[r[[1]]]]]]; Table[Print[a[n]]; a[n], {n, 0, 9}] (* Jean-François Alcover, Jan 08 2013 *)

Crossrefs

Cf. A182121.

Sequence in context: A234647 A106949 A236687 * A217306 A106981 A176985

Adjacent sequences: A187759 A187760 A187761 * A187763 A187764 A187765

Keyword

nonn

Author

Vikram Pandya, Jan 04 2013

Status

approved