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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
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
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Feb 21 2019
STATUS
approved