A006308 Coefficients of period polynomials. (Formerly M2834)

3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050

Offset

2,1

Comments

Conjecture: a(n) = A008837(n) = p*(p-1)/2 = Sum_{k=0..p-1} mod(k^3, p) where p = prime(n). - Michael Somos, Feb 17 2020

References

D. H. and Emma Lehmer, Cyclotomy for nonsquarefree moduli, pp. 276-300 of Analytic Number Theory (Philadelphia 1980), Lect. Notes Math. 899 (1981).

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

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 243.

Links

Table of n, a(n) for n=2..26.

Sean A. Irvine, The polynomials f(p,x) for primes p=3..101

Formula

For an odd prime p, let g be a primitive root of p^2, q=g^p, and zeta=exp(2*pi*i/p^2). Define h(p,k) = Sum_{j=0..p-2} zeta^((q+k*p)*q^j) and a polynomial f(p,x) = Product_{k=0..p-1} (x-h(p,k)). Finally, a(n) = -[x^(p-2)] f(p,x) where p = A000040(n) is the n-th prime. - Sean A. Irvine, Mar 07 2017

Crossrefs

Cf. A008837. [From R. J. Mathar, Oct 28 2008]

Sequence in context: A192033 A295063 A298856 * A008837 A176098 A355389

Adjacent sequences: A006305 A006306 A006307 * A006309 A006310 A006311

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms and offset corrected by Sean A. Irvine, Mar 07 2017

Status

approved