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

1,2

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.

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.

Example

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

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]

Prog

(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

Crossrefs

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

Sequence in context: A076564 A325044 A343822 * A164563 A179892 A348519

Adjacent sequences: A303547 A303548 A303549 * A303551 A303552 A303553

Keyword

nonn,easy

Author

Amiram Eldar, Apr 26 2018

Status

approved