

A079612


Largest number m such that a^n == 1 (mod m) whenever a is coprime to m.


10



2, 24, 2, 240, 2, 504, 2, 480, 2, 264, 2, 65520, 2, 24, 2, 16320, 2, 28728, 2, 13200, 2, 552, 2, 131040, 2, 24, 2, 6960, 2, 171864, 2, 32640, 2, 24, 2, 138181680, 2, 24, 2, 1082400, 2, 151704, 2, 5520, 2, 1128, 2, 4455360, 2, 264, 2, 12720, 2, 86184, 2, 13920
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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. AdamsWatters, 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. 285324 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. 56.
P. J. Cameron and D. A. Preece, Notes on primitive lambdaroots, 2009. See lambda*() in theorem 5.2 (b) p. 8.
Joris van der Hoeven and GrĂ©goire Lecerf, Sparse polynomial interpolation. Exploring fast heuristic algorithms over finite fields, Simon Fraser University (BC Canada) / Institut Polytechnique de Paris (France, 2019) hal02382117.
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 p1 divides n and p^t exactly divides n).
From Antti Karttunen, Dec 19 2018: (Start)
a(n) = A185633(n)*(2A000035(n)).
It also seems that for n > 1, a(n) = 2*A075180(n1). (End)
We have 2*A075180(2n1) = A006863(n) by definition, and A006863(n) = a(2n) by the comments in A006863. Hence a(n) = 2*A075180(n1) for all even n. For all odd n > 1, we have a(n) = 2, which is also equal to 2*A075180(n1). So the formula above is true.  Jianing Song, Apr 05 2021


PROG

(PARI) a(n) = {if (n%2, 2, res = 1; forprime(p=2, n+1, if (!(n % (p1)), t = valuation(n, p); if (p==2, if (t, res *= p^(t+2)), res *= p^(t+1)); ); ); res; ); } \\ Michel Marcus, May 12 2018


CROSSREFS

Cf. A006863 (bisection except for initial term); A059379 (Jordan function).
Cf. A075180, A115000, A115001, A115002, A115003.
Cf. A143407, A143408, A185633, A322315.
Sequence in context: A350257 A270562 A100816 * A329263 A227477 A351850
Adjacent sequences: A079609 A079610 A079611 * A079613 A079614 A079615


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jan 29 2003


EXTENSIONS

Edited by Franklin T. AdamsWatters, 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



