A359586 Inverse Möbius transform of A359581.

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, 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, 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

Offset

1,2

Comments

Multiplicative because A359581 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

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

From Amiram Eldar, Oct 23 2023: (Start)

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.

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

Mathematica

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

Prog

(PARI)
A329697(n) = if(!bitand(n, n-1), 0, 1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A359581(n) = ((-1)^A329697(n));
A359586(n) = sumdiv(n, d, A359581(d));

Crossrefs

Cf. A329697, A359581.

Sequence in context: A141700 A345228 A357887 * A035205 A357761 A284823

Adjacent sequences: A359583 A359584 A359585 * A359587 A359588 A359589

Keyword

nonn,mult

Author

Antti Karttunen, Jan 07 2023

Status

approved