A006308 Coefficients of period polynomials.

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

Author

N. J. A. Sloane

Keywords

• nonn

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

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

Crossrefs

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

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

Extensions

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