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
Keywords
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.
Programs
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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
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