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).

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

Author

Joerg Arndt, Mar 19 2016

Keywords

• nonn

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

• Gheorghe Coserea, Table of n, a(n) for n = 1..50005
• R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.

Crossrefs

Cf. A002322, A002174, A051222, A143407, A270564, A111725.

See also A321713 (number of solutions).

Programs

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 *)

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

Extensions

Corrected and extended by Gheorghe Coserea, Feb 21 2019

Entry revised by N. J. A. Sloane, May 03 2019