A303550 Numbers k such that abs(60*k^2 - 1710*k + 12150) +- 1 are twin primes.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 27, 33, 34, 35, 36, 38, 41, 50, 56, 57, 64, 66, 69, 75, 81, 85, 86, 90, 93, 98, 103, 106, 119, 121, 133, 136, 141, 143, 146, 150, 181, 182, 189, 195, 202, 207, 208, 212, 215, 218, 219, 225

Offset: 1

Author

Amiram Eldar, Apr 26 2018

Keywords

• nonn
• easy

Comments

The formula was discovered by Andrew T. Gazsi in 1961.

The polynomial can also be given as 30*(2*k - 27)*(k - 15). Its value is negative (-30) at k = 14 and 0 and k = 15.

Beiler erroneously claimed that the polynomial generates twin primes for k = 1 to 20.

Examples

			1 is in the sequence since 60*1^2 - 1710*1 + 12150 = 10500 and (10499, 10501) are twin primes.
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed., Dover Publications, Inc., New York, 1966, p. 225.
• Joseph B. Dence and Thomas P. Dence, Elements of the Theory of Numbers, Academic Press, 1999, problem 1.94, p.35.
• Andrew T. Gazsi, A Formula to Generate Prime Pairs, Recreational Mathematics Magazine, edited by Joseph S. Madachy, Issue 6, December 1961, p. 44.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• James Alston Hope Hunter and Joseph S. Madachy, Mathematical Diversions, D. van Nostrand Company, Inc., Princeton, New Jersey, 1963, p. 7.
• Carlos Rivera, Problem 44. Twin-primes producing polynomials race, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A001097, A001359, A006512, A088485, A108897, A124518, A124519, A139404, A185660.

Programs

Maple

filter:= proc(n) local k;
  k:= abs(60*n^2-1710*n+12150);
  isprime(k+1) and isprime(k-1)
end proc:
select(filter, [$1..300]); # Robert Israel, Jun 19 2018

Mathematica

f[n_] := 60n^2 - 1710n + 12150; aQ[n_]:=PrimeQ[f[n]-1] && PrimeQ[f[n]+1]; Select[Range[225], aQ]
Select[Range[250],AllTrue[Abs[60#^2-1710#+12150+{1,-1}],PrimeQ]&] (* Harvey P. Dale, May 17 2025 *)

PARI

f(n) = abs(60*n^2 - 1710*n + 12150);
isok(n) = my(fn=f(n)); isprime(fn-1) && isprime(fn+1); \\ Michel Marcus, Apr 27 2018