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 A157185 a(n) = least prime p such that there are n consecutive even numbers E < 2 sqrt(p) such that E^2-p is prime. 0
 2, 13, 41, 83, 83, 167, 227, 227, 2273, 2273, 2273, 5297, 340007, 837077, 837077, 837077, 837077, 837077, 2004917, 2004917, 2004917, 208305767, 208305767 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Inspired by a puzzle proposed by J. M. Bergot, cf. link. The limit of 2*sqrt(p) imposed here is somewhat arbitrary, but this seemed the most natural choice. (Note that the E's must be > sqrt(p).) Changing it to 3*sqrt(p) would "reveal" the numbers 5,17 and the fact that p=227 is "good" up to n=13. Beyond a(14) both of these limits yield the same terms. a(24) > 3*10^9. - Donovan Johnson, Nov 29 2010 LINKS J. M. Bergot, Puzzle #481: E.E-p=q, in C. Rivera's Prime puzzles, Feb. 2009. EXAMPLE The following table gives n (length of the run), a(n) (the prime), and the largest of the n consecutive even numbers E,...,E[n] such that E[i]^2-p is prime: [1, 2, 2] since 2^2-2 = 2 is prime [2, 13, 6] since 4^2-5 = 11 and 6^2-5 = 31 are prime [3, 41, 12] [4, 83, 16] is a truncation of the following chain: [5, 83, 18] since {10,12,14,16,18} squared minus 83 are all prime [6, 167, 24] since 20^2-83 which is prime is ignored because 20 > 2sqrt(83) [7, 227, 28] [8, 227, 30] [9, 2237, 64] since E=32,34,36,38,40 are > 2sqrt(227), [10, 2273, 66] although they would also yield primes E^2-p for p=227. [11, 2273, 68] [12, 5297, 96] [13, 340007, 690] [14, 837077, 942] [15, 837077, 944] [16, 837077, 946] [17, 837077, 948] [18, 837077, 950] [19, 2004917, 2572] [20, 2004917, 2574] [21, 2004917, 2576] PROG (PARI) list_A157185( m=0, p=0, C=2, L=0 )={ until( L=0, forstep( j=(t=sqrt(p=nextprime(p+2)))\2*2+2, C*t\1, 2, if(isprime(j^2-p), L++>m & print([m=L, p, j]), L&L=0)))} CROSSREFS Sequence in context: A042795 A179925 A263980 * A219054 A154354 A138089 Adjacent sequences: A157182 A157183 A157184 * A157186 A157187 A157188 KEYWORD more,nonn AUTHOR M. F. Hasler, Mar 02 2009 EXTENSIONS a(22)-a(23) from Donovan Johnson, Nov 29 2010 STATUS approved

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Last modified March 22 09:08 EDT 2023. Contains 361423 sequences. (Running on oeis4.)