OFFSET
1,1
COMMENTS
a(m) divides the Jordan function J_m(n) for all n except when n is a prime dividing a(m) or m=2, n=4; it is the largest number dividing all but finitely many values of J_m(n). For m > 0, a(m) also divides Sum_{k=1}^n J_m(k) for n >= the largest exceptional value. - Franklin T. Adams-Watters, Dec 10 2005
The numbers m with this property are the divisors of a(n) that are not divisors of a(r) for r<n.
REFERENCES
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003. (The function K(n), see p. 303.)
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
Shigeki Akiyama and Hajime Kaneko, Curious congruences on cyclotomic polynomials, arXiv:2204.11267 [math.NT], 2022. See Proposition 1 p. 5-6.
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots, 2009. See lambda*() in theorem 5.2 (b) p. 8.
Joris van der Hoeven and Grégoire Lecerf, Sparse polynomial interpolation. Exploring fast heuristic algorithms over finite fields, Simon Fraser University (BC Canada) / Institut Polytechnique de Paris (France, 2019) hal-02382117.
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, The function K(n), see p. 19.
FORMULA
a(n) = 2 for n odd; for n even, a(n) = product of 2^(t+2) (where 2^t exactly divides n) and p^(t+1) (where p runs through all odd primes such that p-1 divides n and p^t exactly divides n).
From Antti Karttunen, Dec 19 2018: (Start)
It also seems that for n > 1, a(n) = 2*A075180(n-1). (End)
PROG
(PARI) a(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p-1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)); ); ); res; ); } \\ Michel Marcus, May 12 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 29 2003
EXTENSIONS
Edited by Franklin T. Adams-Watters, Dec 10 2005
Definition corrected by T. D. Noe, Aug 13 2008
Rather arbitrary term a(0) removed by Max Alekseyev, May 27 2010
STATUS
approved