A321713 -id:A321713 - cp's OEIS frontend

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

Joerg Arndt, Mar 19 2016

nonn

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.

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.

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

See also A321713 (number of solutions).

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 *)

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

Corrected and extended by Gheorghe Coserea, Feb 21 2019

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

A321714 Numbers k such that lambda(k) = 12.

Original entry on oeis.org

13, 26, 35, 39, 45, 52, 65, 70, 78, 90, 91, 104, 105, 112, 117, 130, 140, 144, 156, 180, 182, 195, 208, 210, 234, 260, 273, 280, 312, 315, 336, 360, 364, 390, 420, 455, 468, 520, 546, 560, 585, 624, 630, 720, 728, 780, 819, 840, 910, 936, 1008, 1040, 1092, 1170, 1260, 1365, 1456, 1560, 1638, 1680, 1820, 1872, 2184, 2340, 2520, 2730, 3120, 3276, 3640, 4095, 4368, 4680, 5040, 5460, 6552, 7280, 8190, 9360, 10920, 13104, 16380, 21840, 32760, 65520

Offset: 1

Gheorghe Coserea, Feb 21 2019

Here lambda is Carmichael's lambda function (see A002322).

R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1910), 232-238.

Cf. A002322, A270562, A321713.

Select[Range[65520], CarmichaelLambda[#] == 12 &] (* Paolo Xausa, Feb 28 2024 *)

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))));
};
invlambda(n) = { \\ A270562
  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);
};
lambda_level(n) = {
  my(N = invlambda(n)); if (!N, return([])); my(s=List());
  fordiv(N, d, if (lambda(d) == n, listput(s, d)));
  Set(s);
};
lambda_level(12)

cp's OEIS Frontend

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).

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