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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 (list; graph; refs; listen; history; text; internal format)
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

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.

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-Fran├žois Alcover, Oct 18 2019, after Gheorghe Coserea *)

PROG

(PARI, from Gheorghe Coserea, Feb 21 2019)

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))

CROSSREFS

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

See also A321713 (number of solutions).

Sequence in context: A228241 A054909 A171636 * A100816 A079612 A329263

Adjacent sequences:  A270559 A270560 A270561 * A270563 A270564 A270565

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

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Last modified January 15 10:12 EST 2021. Contains 340187 sequences. (Running on oeis4.)