A359588 Dirichlet inverse of A083346.

1, -2, -3, 3, -5, 6, -7, -6, 6, 10, -11, -9, -13, 14, 15, 12, -17, -12, -19, -15, 21, 22, -23, 18, 20, 26, -10, -21, -29, -30, -31, -24, 33, 34, 35, 18, -37, 38, 39, 30, -41, -42, -43, -33, -30, 46, -47, -36, 42, -40, 51, -39, -53, 20, 55, 42, 57, 58, -59, 45, -61, 62, -42, 48, 65, -66, -67, -51, 69

Offset: 1

Author

Antti Karttunen, Jan 09 2023

Keywords

• sign
• mult
• look

Comments

Multiplicative because A083346 is.

Links

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

Crossrefs

Cf. A083346, A091428 (positions of odd terms), A359592 (parity of terms).

Cf. also A359577.

Programs

Mathematica

f[p_, e_] := If[Divisible[e, p], 1, p]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, May 18 2023 *)

PARI

A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
memoA359588 = Map();
A359588(n) = if(1==n,1,my(v); if(mapisdefined(memoA359588,n,&v), v, v = -sumdiv(n,d,if(dA083346(n/d)*A359588(d),0)); mapput(memoA359588,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083346(n/d) * a(d).