A001912 -id:A001912 - cp's OEIS frontend

A334294 Numbers k such that 70k^2 + 70k - 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

James R. Buddenhagen, Apr 21 2020

nonn

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.

			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.

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.

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

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

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);

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

A366193 For n >= 0, a(n) is the least x >= 0 such that x^2 + (x + 2*n)^2 + 1 = p, p prime number (A000040).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 9, 0, 0, 6, 0, 6, 0, 0, 3, 15, 1, 2, 0, 1, 0, 6, 1, 2, 6, 3, 9, 0, 0, 6, 15, 4, 5, 0, 3, 2, 6, 0, 2, 3, 1, 9, 0, 4, 3, 0, 7, 0, 3, 1, 6, 6, 1, 5, 6, 0, 2, 6, 0, 6, 0, 1, 0, 0, 13, 0, 6, 0, 6, 3, 4, 11, 12, 0, 3, 0, 9, 3, 0, 3, 0, 21, 9, 2, 3, 0, 6, 18, 0, 3

Offset: 0

Ctibor O. Zizka, Oct 03 2023

nonn

For a(n) = 0 the resulting primes p >= 5 see in A002496.

			n = 0: x^2 + x^2 + 1 = p is valid for the least x = 1, p = 3, thus a(0) = 1.
n = 6: x^2 + (x + 12)^2 + 1 = p is valid for the least x = 9, p = 523, thus a(6) = 9.

Stoyan I. Dimitrov, A ternary diophantine inequality by primes with one of the form p = x^2 + y^2 + 1, arXiv:2011.03967 [math.NT], 2020.

Cf. A000040, A001912, A002496.

a(n) = my(x=0); while (!isprime(x^2 + (x + 2*n)^2 + 1), x++); x; \\ Michel Marcus, Oct 03 2023

a(n) = 0 for n from A001912.

More terms from Michel Marcus, Oct 03 2023

A386516 Least k such that k^2+1 contains exactly n distinct prime factors of the form m^2+1 or 0 if no such k exists.

Original entry on oeis.org

1, 3, 13, 183, 2843, 41323, 57109753, 1929510527, 760999599793

Offset: 1

Michel Lagneau, Jul 24 2025

			a(4)=183 because the prime factors of 183^2+1 are {2, 5, 17, 197} are of the form m^2+1 with m = 1, 2, 4 and 14.

Cf. A002522, A001912, A002496, A005574, A008784, A216784.

with(numtheory):nn:=10^6:
for n from 0 to 6 do :
 ii :=0 :
 for k from 0 to nn while(ii=0) do :
   d:=factorset(k^2+1):n0:=nops(d):it:=0:
    for i from 1 to n0 do:
      c:=d[i]-1:if sqrt(c) = floor(sqrt(c)) then it:=it+1:else fi:
    od:
     if it =n then ii :=1 :printf (`%d %d \n`,n,k):
      else
     fi :
 od:
od :

a[n_]:=Module[{k=0},Until[PrimeNu[k^2+1]==n&&AllTrue[Sqrt[First/@FactorInteger[k^2+1]-1],IntegerQ],k++];k];Array[a,6] (* James C. McMahon, Jul 25 2025 *)

a(7) from Giovanni Resta, Jul 24 2025

a(8)-a(9) from David A. Corneth, Jul 24 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A366193 For n >= 0, a(n) is the least x >= 0 such that x^2 + (x + 2*n)^2 + 1 = p, p prime number (A000040).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A386516 Least k such that k^2+1 contains exactly n distinct prime factors of the form m^2+1 or 0 if no such k exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A366193 For n >= 0, a(n) is the least x >= 0 such that x^2 + (x + 2*n)^2 + 1 = p, p prime number (A000040).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A386516 Least k such that k^2+1 contains exactly n distinct prime factors of the form m^2+1 or 0 if no such k exists.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

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