A334294 Numbers k such that 70*k^2 + 70*k - 1 is prime.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 46, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 74, 76, 77, 78, 79, 80, 81, 82, 87, 88, 90, 93, 96, 97, 100

Offset

1,2

Comments

Among quadratic polynomials in k of the form a*k^2 + a*k - 1 the value a=70 gives the most primes for any a in the range 1<=a<=300, at least up to k=40000. Here a and k are positive integers. Other "good" values of a are a=250, a=99, and a=19.

Links

Table of n, a(n) for n=1..63.

N. Boston et M. L.Greenwood, Quadratics representing primes, Amer. Math. Monthly 102:7 (1995), 595-599.

François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.

G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.

R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Example

For k=1, 70*k^2 + 70*k - 1 = 70*1^2 + 70*1 - 1 = 139, which is prime, so 1 is in the sequence.

Maple

a:=proc(n) if isprime(70*n^2+70*n-1) then n else NULL end if end proc;
seq(a(n), n=1..100);

Mathematica

Select[Range@ 100, PrimeQ[70 #^2 + 70 # - 1] &] (* Michael De Vlieger, May 26 2020 *)

Crossrefs

Cf. A001912, A002384, A005574, A027861, A028870, A056900.

Cf. A067201, A088572, A089623, A091199, A271980.

Sequence in context: A368904 A229991 A178156 * A177054 A247760 A247750

Adjacent sequences: A334291 A334292 A334293 * A334295 A334296 A334297

Keyword

nonn

Author

James R. Buddenhagen, Apr 21 2020

Status

approved