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.

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

Offset

1,1

Links

Bertram Felgenhauer, Table of n, a(n) for n = 1..10000 (first 3023 terms by Gheorghe Coserea)

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))
(PARI)
b(n) = { \\ number of k satisfying lambda(k) | n
my(R = 1);
fordiv (n, d, if(isprime(d+1),
  my(e = 1); while(n % (d+1) == 0, n /= d+1; e++);
  if (d == 1 && e > 1, e++); R *= e+1));
R
};
a(n) = if (n <= 0, 0, n == 1, 2, n % 2, 0, sumdiv(n, d, moebius(n/d) * b(d)));
vector(100, n, a(n)) \\ Bertram Felgenhauer, Mar 27 2022

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