A382941 a(n) = exp(Sum_{d|n} A382883(d)*log(n/d)).

1, 2, 3, 2, 5, 1, 7, 4, 3, 1, 11, 3, 13, 1, 1, 16, 17, 2, 19, 5, 1, 1, 23, 18, 5, 1, 9, 7, 29, 1, 31, 64, 1, 1, 1, 36, 37, 1, 1, 50, 41, 1, 43, 11, 5, 1, 47, 72, 7, 2, 1, 13, 53, 12, 1, 98, 1, 1, 59, 15, 61, 1, 7, 512, 1, 1, 67, 17, 1, 1, 71, 648, 73, 1, 3, 19

Offset: 1

Author

Peter Luschny, Apr 09 2025

Keywords

• nonn

Comments

See the comments in A382883.

Crossrefs

Cf. A000469, A014963, A205959, A382883, A383263.

Programs

Maple

h := proc(n) option remember; local j; ifelse(n = 1, 1,
-add(ifelse(j = 1, 1, padic:-ordp(n, j))*h(j), j = 1..n-1)) end:
a := n -> local d; simplify(exp(add(h(d)*log(n/d), d in numtheory:-divisors(n)))):
seq(a(n), n = 1..76);

Mathematica

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; a[n_] := Exp[DivisorSum[n, f[#] * Log[n/#] &]]; Array[a, 100] (* Amiram Eldar, Apr 29 2025 *)

SageMath

def A382941(n: int) -> int: return simplify(exp(sum(A382883(d)*log(n//d) for d in n.divisors())))
print([A382941(n) for n in srange(1, 76)]);

Formula

Restricting the sum to prime divisors of n gives A205959(n).

a(k) = 1 <=> k in A000469 <=> k is a nonprime squarefree number.

a(k) != 1 <=> k in A383263 <=> k is prime or divisible by a square greater than 1.