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A270562
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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).
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Joerg Arndt, Mar 19 2016
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EXTENSIONS
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Corrected and extended by Gheorghe Coserea, Feb 21 2019
Entry revised by N. J. A. Sloane, May 03 2019
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STATUS
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approved
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