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.)
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
KEYWORD
nonn
AUTHOR
Vikram Pandya, Jan 04 2013
STATUS
approved