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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Apr 26 2018

Keywords

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

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

A317029 Invertible primes p such that k*p - 1 and k*p + 1 is a twin prime pair; for k = 12.

Original entry on oeis.org

19, 601, 1601, 16661, 16981, 19609, 60689, 66809, 69001, 69011, 100169, 119191, 189901, 196919, 616961, 1061689, 1088089, 1091119, 1106069, 1196089, 1198069, 1611601, 1666019, 1688969, 1800119, 1861889, 1891619, 1891661, 1910669, 1996681, 6060091, 6160601, 6196909
Offset: 1

Views

Author

K. D. Bajpai, Jul 19 2018

Keywords

Comments

Intersection of A048890 (invertible primes) and A138242.
k = 12 is the smallest integer to produce such sequence.

Examples

			a(2) = 601 is an invertible prime; 12*601 - 1 = 7211; 12*601 + 1 = 7213; 7211 and 7213 form a twin prime pair.
a(4) = 16661 is an invertible prime; 12*16661 - 1 = 199931; 12*16661 + 1 = 199933; 199931 and 199933 form a twin prime pair.

Crossrefs

Cf. A048890, A001359, A006512, A124519, A138242.

Programs

  • Mathematica
    k = 12; Select[lst = {};
fQ[n_] := Block[{allset = {0, 1, 6, 8, 9}, id = IntegerDigits@n}, rid = Reverse[id /. {6 -> 9, 9 -> 6}];Union@Join[id, allset] == allset && PrimeQ@FromDigits@rid && rid != id];Do[If[PrimeQ@n && fQ@n, AppendTo[lst, n]], {n, 12000000}]; lst,
PrimeQ[k# + 1] && PrimeQ[k# - 1] &]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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