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 A214089 Least prime p such that the first n primes divide p^2-1. 7
 3, 5, 11, 29, 419, 1429, 1429, 315589, 1729001, 57762431, 1724478911, 6188402219, 349152569039, 1430083494841, 390499187164241, 1010518715554349, 18628320726623609, 522124211958421799, 522124211958421799, 5936798290039408015951, 311263131154464891496249 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (a(n)^2 - 1) / A002110(n) is congruent to 0 (mod 4). (a(n)^2 - 1) / (4*A002110(n)) = A215085(n). [J. Stauduhar, Aug 03 2012] a(n) == +1 or -1 (mod prime(i)) for every i=1,2,...,n. The system of congruences x == +1 or -1 (mod prime(i)), i=1,2,...,n, has 2^(n-1) solutions modulo A002110(n) so that a(n) represents the smallest prime in the corresponding residue classes, allowing efficient computation (see PARI program). - Max Alekseyev, Aug 22 2012 LINKS Max Alekseyev, Table of n, a(n) for n = 1..32 EXAMPLE a(5) = 419: 419^2-1 = 175560 = 2^3*3*5*7*11*19 contains the first 5 primes. a(7) = 1429: 1428=2^2*3*7*17, 1430=2*5*11*13 contains the first 7 primes. a(8) = 315589: 315589^2-1 = 2^3*3*5*7*11*13*17^2*19*151 contains the first 8 primes. MAPLE A214089 := proc(n) local m, k, p; m:= 2*mul(ithprime(j), j=1..n); for k from 1 do p:= sqrt(m*k+1); if type(p, integer) and isprime(p) then return(p) end if end do end proc; # Robert Israel, Aug 19 2012 MATHEMATICA f[n_] := Block[{k = 1, p = Times @@ Prime@Range@n}, While[! IntegerQ@Sqrt[4 k*p + 1], k++]; Block[{j = k}, While[! PrimeQ[Sqrt[4 j*p + 1]], j++]; Sqrt[4 j*p + 1]]]; Array[f, 10] (* J. Stauduhar, Aug 18 2012 *) PROG (PARI) A214089(n) = { local(a, k=4, p) ; a=prod(j=1, n, prime(j)) ; while(1, if( issquare(k*a+1, &p), if(isprime(p), return(p); ) ; ) ; k+=4; ) ; } ; (PARI) { a(n) = local(B, q); B=prod(i=1, n, prime(i))^2; forvec(v=vector(n-1, i, [0, 1]), q=chinese(concat(vector(n-1, i, Mod((-1)^v[i], prime(i+1))), [Mod(1, 2)])); forstep(s=lift(q), B-1, q.mod, if(ispseudoprime(s), B=s; break)) ); B } /* Max Alekseyev, Aug 22 2012 */ (Python) from itertools import product from sympy import sieve, prime, isprime from sympy.ntheory.modular import crt def A214089(n): return 3 if n == 1 else int(min(filter(isprime, (crt(tuple(sieve.primerange(prime(n)+1)), t)[0] for t in product((1, -1), repeat=n))))) # Chai Wah Wu, May 31 2022 CROSSREFS Cf. A073917, A103783. Sequence in context: A265784 A146243 A262936 * A108259 A093933 A165572 Adjacent sequences: A214086 A214087 A214088 * A214090 A214091 A214092 KEYWORD nonn,hard AUTHOR Robin Garcia, Jul 02 2012 EXTENSIONS a(15)-a(16) from Donovan Johnson, Jul 25 2012 a(17) from Charles R Greathouse IV, Aug 08 2012 a(18) from Charles R Greathouse IV, Aug 16 2012 a(19) from J. Stauduhar, Aug 18 2012 a(20)-a(32) from Max Alekseyev, Aug 22 2012 STATUS approved

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Last modified September 11 01:27 EDT 2024. Contains 375813 sequences. (Running on oeis4.)