A369008 a(n) = A085731(n) / A003557(n).

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Jan 15 2024

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A003415, A003557, A083345, A085731, A342001, A369009.

Cf. A342090 (positions of terms > 1).

Programs

Mathematica

f[p_, e_] := If[Divisible[e, p], p, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2024 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A369008(n) = { my(u=A003415(n)); (gcd(n,u)/A003557(n)); };

PARI

A369008(n) = if(1==n, n, my(f=factor(n)); for(i=1, #f~, if((f[i, 2]%f[i, 1]), f[i, 1] = 1, f[i, 2] = 1)); factorback(f));

Formula

Multiplicative with a(p^e) = p if p|e, otherwise a(p^e) = 1.

For n > 1, a(n) = A342001(n) / A083345(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^(p+1) + p^2 - 3*p +1)/(p*(p^p-1))) = 1.22775972725472961826... . - Amiram Eldar, Jan 20 2024