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A321713 a(n) is the number of values k satisfying lambda(k)=n or zero if there is no solution, where lambda(k) is Carmichael's lambda function. 3
2, 6, 0, 12, 0, 16, 0, 4, 0, 8, 0, 84, 0, 0, 0, 32, 0, 40, 0, 32, 0, 8, 0, 20, 0, 0, 0, 20, 0, 64, 0, 8, 0, 0, 0, 480, 0, 0, 0, 80, 0, 48, 0, 12, 0, 8, 0, 160, 0, 0, 0, 20, 0, 16, 0, 4, 0, 8, 0, 1216, 0, 0, 0, 8, 0, 64, 0, 0, 0, 16, 0, 872, 0, 0, 0, 0, 0, 24, 0, 160, 0, 8, 0, 532, 0, 0, 0, 52, 0, 120, 0, 12, 0, 0, 0, 424, 0, 0, 0, 100 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 1..3023

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

EXAMPLE

For n=12 there are a(12)=84 values N satisfying lambda(N)=12; the values are enumerated in A321714.

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))));

};

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);

};

a(n) = length(lambda_level(n));

vector(100, n, a(n))

CROSSREFS

Cf. A002322, A270562, A321714.

Sequence in context: A180314 A065344 A131105 * A057635 A269943 A243015

Adjacent sequences:  A321710 A321711 A321712 * A321714 A321715 A321716

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Feb 21 2019

STATUS

approved

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Last modified January 19 21:47 EST 2020. Contains 331066 sequences. (Running on oeis4.)