A000921 Primes p of the form 3k+1 such that Sum_{x=1..p} cos(2*Pi*x^3/p) > sqrt(p). (Formerly M4398 N1854)

7, 31, 43, 67, 73, 79, 103, 127, 163, 181, 223, 229, 271, 277, 307, 313, 337, 349, 409, 421, 439, 457, 463, 499, 523, 577, 643, 661, 673, 691, 709, 727, 757, 769, 811, 823, 829, 853, 877, 919, 967, 991, 997, 1021, 1069, 1087, 1093, 1117, 1123, 1171, 1213

Offset

1,1

Comments

For the first 1000 terms in this sequence (primes up to 44683), the minimum difference between sqrt(p) and the sum is 1.47633.... Hence there does not seem to be a need to compute the sum to high precision. - T. D. Noe, Jun 20 2012

References

H. Hasse, Vorlesungen über Zahlentheorie. Springer-Verlag, NY, 1964, p. 482.

G. B. Mathews, Theory of Numbers, 2nd edition. Chelsea, NY, p. 228.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

D. R. Heath-Brown, Kummer's Conjecture for Cubic Gauss Sums

J. von Neumann and H. H. Goldstine, A numerical study of a conjecture of Kummer, Math. Comp., 7 (1953), 133-134.

J. von Neumann and H. H. Goldstine, A numerical study of a conjecture of Kummer, Math. Comp., 7 (1953), 133-134. [Annotated scanned copy]

Example

7 is here because the sum of cos(2*Pi*x^3/7) = 4.7409 > sqrt(7).

Mathematica

isok[p_] :=Mod[p, 3]==1 && PrimeQ[p] && Sum[Cos[2*Pi*x^3/p], {x, 1, p}] > Sqrt[p]; Select[Range[1213], isok] (* James C. McMahon, Dec 10 2023 *)

Prog

(PARI) isok(p) = isprime(p) && ((p % 3) == 1) && (sum(x=1, p, cos(2*Pi*x^3/p)) > sqrt(p)); \\ Michel Marcus, Oct 16 2017

Crossrefs

Cf. A000922, A000923, A002476.

Sequence in context: A000696 A171733 A128028 * A185004 A172490 A349666

Adjacent sequences: A000918 A000919 A000920 * A000922 A000923 A000924

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Don Reble, May 26 2007

Status

approved