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

Author

Gheorghe Coserea, Feb 21 2019

Keywords

• nonn

Examples

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

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.

Crossrefs

Cf. A002322, A270562, A321714.

Programs

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