A000921 - OEIS

%I M4398 N1854 #42 Dec 12 2023 08:22:45

%S 7,31,43,67,73,79,103,127,163,181,223,229,271,277,307,313,337,349,409,

%T 421,439,457,463,499,523,577,643,661,673,691,709,727,757,769,811,823,

%U 829,853,877,919,967,991,997,1021,1069,1087,1093,1117,1123,1171,1213

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

%C 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

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

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

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

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

%H T. D. Noe, <a href="/A000921/b000921.txt">Table of n, a(n) for n = 1..1000</a>

%H D. R. Heath-Brown, <a href="http://eprints.maths.ox.ac.uk/158/01/kummer.pdf">Kummer's Conjecture for Cubic Gauss Sums</a>

%H J. von Neumann and H. H. Goldstine, <a href="https://doi.org/10.1090/S0025-5718-1953-0055784-0">A numerical study of a conjecture of Kummer</a>, Math. Comp., 7 (1953), 133-134.

%H J. von Neumann and H. H. Goldstine, <a href="/A000921/a000921.pdf"> A numerical study of a conjecture of Kummer</a>, Math. Comp., 7 (1953), 133-134. [Annotated scanned copy]

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

%t 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 *)

%o (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

%Y Cf. A000922, A000923, A002476.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Edited by _Don Reble_, May 26 2007