A159829 a(n) is the smallest natural number m such that n^3+m^3+1^3 is prime.

1, 2, 1, 2, 1, 4, 15, 2, 3, 2, 11, 10, 9, 2, 7, 14, 5, 4, 9, 2, 15, 2, 7, 16, 15, 8, 13, 2, 1, 10, 3, 4, 15, 2, 11, 10, 9, 2, 7, 6, 13, 22, 5, 2, 1, 6, 29, 10, 29, 10, 3, 2, 11, 12, 3, 8, 3, 2, 19, 6, 15, 8, 1, 2, 1, 18, 5, 2, 1, 18, 1, 12, 17, 14, 15, 26, 7, 6, 3, 2, 19, 12, 1, 18, 3, 8, 15, 2, 11, 6

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009

Keywords

• nonn

Comments

a(2k-1) is odd, a(2k) is even.

Exponent 2: There are infinitely many primes of the forms n^2+m^2 and n^2+m^2+1^2.

Exponent k>2: Are there infinitely many primes of the forms n^k+m^k and n^k+m^k+1^k?

Examples

			2^3+2^3+1=17 = A000040(7); a(2)=2.
7^3+15^3+1=3719 = A000040(519); a(7)=15.
21^3+15^3+1=18523 = A000040(2122), a(21)=15.
		

References

• L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999.
• A. Weil, Number theory: an approach through history, Birkhäuser 1984.
• David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005.

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A069003, A159828.

Cf. A067200 (when m=1).

Programs

Maple

A159829 := proc(n) for m from 1 do if isprime(n^3+m^3+1) then RETURN(m) ; fi; od: end: seq(A159829(n),n=1..120) ; # R. J. Mathar, Apr 28 2009

Mathematica

snn[n_]:=Module[{n3=n^3,m=1},While[!PrimeQ[n3+1+m^3],m++];m]; Array[ snn,100] (* Harvey P. Dale, Sep 04 2019 *)

PARI

a(n) = my(m=1); while (!isprime(n^3+m^3+1^3), m++); m; \\ Michel Marcus, Nov 07 2023

Extensions

Corrected and extended by R. J. Mathar, Apr 28 2009