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A270562
a(n) is the largest number m satisfying lambda(m)=n, or zero if there is no solution, where lambda(m) is Carmichael's lambda function A002322(m).
4
2, 24, 0, 240, 0, 504, 0, 480, 0, 264, 0, 65520, 0, 0, 0, 16320, 0, 28728, 0, 13200, 0, 552, 0, 131040, 0, 0, 0, 6960, 0, 171864, 0, 32640, 0, 0, 0, 138181680, 0, 0, 0, 1082400, 0, 151704, 0, 5520, 0, 1128, 0, 4455360, 0, 0, 0, 12720, 0, 86184, 0, 13920, 0, 1416, 0, 6814407600, 0, 0, 0, 65280
OFFSET
1,1
COMMENTS
a(n) is the largest modulus m such that the largest order of any element in the multiplicative group modulo m is n; a(n) is zero if there is no such group with largest order n.
Omitting the zeros gives A143407.
a(n) = 0 if n is not a term of A002174.
LINKS
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.
MATHEMATICA
a[n_] := Module[{f, fsz, g = 1, h = 1, p, e}, Which[n <= 0, Return[0], n == 1, Return[2], OddQ[n], Return[0]]; f = FactorInteger[n][[All, 1]]; fsz = Length[f]; For[k = 1, k <= fsz, k++, p = f[[k]]; e = 1; While[Mod[n, CarmichaelLambda[p^e]] == 0, e++]; g *= p^(e-1)]; Do[If[PrimeQ[d+1] && Mod[g, d+1] != 0, h *= (d+1)], {d, Divisors[n]}]; g *= h; If[CarmichaelLambda[g] != n, 0, g]];
a /@ Range[100] (* Jean-François Alcover, Oct 18 2019, after Gheorghe Coserea *)
PROG
(PARI)
lambda(n) = { \\ A002322
my(f=factor(n), fsz=matsize(f)[1]);
lcm(vector(fsz, k, my(p=f[k, 1], e=f[k, 2]);
if (p != 2, p^(e-1)*(p-1), e > 2, 2^(e-2), 2^(e-1))));
};
a(n) = {
if (n <= 0, return(0), n==1, return(2), n%2, return(0));
my(f=factor(n), fsz=matsize(f)[1], g=1, h=1);
for (k=1, fsz, my(p=f[k, 1], e=1);
while (n % lambda(p^e) == 0, e++); g *= p^(e-1));
fordiv(n, d, if (isprime(d+1) && g % (d+1) != 0, h *= (d+1)));
g *= h; if (lambda(g) != n, 0, g);
};
vector(64, n, a(n)) \\ Gheorghe Coserea, Feb 21 2019
CROSSREFS
See also A321713 (number of solutions).
Sequence in context: A171636 A375304 A350257 * A100816 A079612 A329263
KEYWORD
nonn
AUTHOR
Joerg Arndt, Mar 19 2016
EXTENSIONS
Corrected and extended by Gheorghe Coserea, Feb 21 2019
Entry revised by N. J. A. Sloane, May 03 2019
STATUS
approved