

A303550


Numbers n such that abs(60n^2  1710n + 12150) + 1 are twin primes.


1



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
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OFFSET

1,2


COMMENTS

The formula was discovered by Andrew T. Gazsi in 1961.
The polynomial can also be given as 30(2n  27)(n  15). Its value is negative (30) at n = 14 and 0 and n = 15.
Beiler erroneously claimed that the polynomial generates twin primes for n = 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. Twinprimes 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^21710*n+12150);
isprime(k+1) and isprime(k1)
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(fn1) && isprime(fn+1); \\ Michel Marcus, Apr 27 2018


CROSSREFS

Cf. A001097, A001359, A006512, A088485, A108897, A124518, A124519, A139404, A185660.
Sequence in context: A070776 A320230 A076564 * A179892 A061773 A125007
Adjacent sequences: A303547 A303548 A303549 * A303551 A303552 A303553


KEYWORD

nonn,easy


AUTHOR

Amiram Eldar, Apr 26 2018


STATUS

approved



