A359586 - OEIS

%I #14 Apr 25 2024 09:33:57

%S 1,2,0,3,0,0,2,4,1,0,2,0,2,4,0,5,0,2,0,0,0,4,0,0,1,4,0,6,0,0,0,6,0,0,

%T 0,3,0,0,0,0,2,0,2,6,0,0,2,0,3,2,0,6,0,0,0,8,0,0,2,0,0,0,2,7,0,0,2,0,

%U 0,0,2,4,0,0,0,0,4,0,2,0,1,4,0,0,0,4,0,8,0,0,4,0,0,4,0,0,2,6,2,3,0,0,0,8,0

%N Inverse Möbius transform of A359581.

%C Multiplicative because A359581 is.

%H Antti Karttunen, <a href="/A359586/b359586.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = Sum_{d|n} A359581(d).

%F From _Amiram Eldar_, Oct 23 2023: (Start)

%F Multiplicative with a(p^e) = e+1 if A329697(p) is even, and if A329697(p) is odd, 0 if e is odd and 1 if e is even.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} p/(p-(-1)^A329697(p)) = 1.4128... . (End)

%t A329697[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] - 1;

%t f[p_, e_] := If[EvenQ[A329697[p]], e + 1, If[OddQ[e], 0, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 23 2023 *)

%o (PARI)

%o A329697(n) = if(!bitand(n, n-1), 0, 1+A329697(n-(n/vecmax(factor(n)[, 1]))));

%o A359581(n) = ((-1)^A329697(n));

%o A359586(n) = sumdiv(n,d,A359581(d));

%Y Cf. A329697, A359581.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Jan 07 2023