

A079612


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


7



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
P. J. Cameron and D. A. Preece, Notes on primitive lambdaroots, 2009. See lambda*() in theorem 5.2 (b) p. 8.
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)


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: A171636 A270562 A100816 * A227477 A066585 A278563
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



