A159828 -id:A159828 - cp's OEIS frontend

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

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

nonn

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?

			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.

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.

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

Cf. A069003, A159828.

Cf. A067200 (when m=1).

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

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 *)

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

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

A176983 Primes p such that smallest prime q > p^2 is of form q = p^2 + k^2.

Original entry on oeis.org

2, 5, 7, 13, 17, 37, 47, 67, 73, 97, 103, 137, 163, 167, 193, 233, 277, 281, 293, 307, 313, 317, 347, 373, 389, 421, 439, 461, 463, 487, 499, 503, 547, 571, 577, 593, 607, 613, 661, 677, 691, 701, 739, 743, 769, 787, 821, 823, 827, 829, 853, 883, 953, 967, 983

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 30 2010

nonn

By Fermat's 4n+1 theorem, q must be congruent to 1 (mod 4).

Corresponding values of k: 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 4, 6, 2, 2, 4, 2. - Zak Seidov, Nov 04 2013

			17 is here because 293 is the first prime after 17^2 and 293 = 17^2 + 2^2.

Eric W. Weisstein, Fermat's 4n+1 Theorem

Cf. A000040, A000290, A002144, A159828.

A062324 is subsequence. - Zak Seidov, Nov 04 2013

Select[Prime[Range[200]], IntegerQ[Sqrt[NextPrime[ #^2] - #^2]] & ]

Edited and extended by T. D. Noe, May 12 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions