cp's OEIS Frontend

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

%I A369008 #15 Jan 20 2024 03:16:22
%S A369008 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,
%T A369008 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,
%U A369008 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
%N A369008 a(n) = A085731(n) / A003557(n).
%H A369008 Antti Karttunen, <a href="/A369008/b369008.txt">Table of n, a(n) for n = 1..100000</a>
%F A369008 Multiplicative with a(p^e) = p if p|e, otherwise a(p^e) = 1.
%F A369008 For n > 1, a(n) = A342001(n) / A083345(n).
%F A369008 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
%t A369008 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 *)
%o A369008 (PARI)
%o A369008 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369008 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A369008 A369008(n) = { my(u=A003415(n)); (gcd(n,u)/A003557(n)); };
%o A369008 (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));
%Y A369008 Cf. A003415, A003557, A083345, A085731, A342001, A369009.
%Y A369008 Cf. A342090 (positions of terms > 1).
%K A369008 nonn,mult
%O A369008 1,4
%A A369008 _Antti Karttunen_, Jan 15 2024