A359432 Dirichlet inverse of A327936, which is multiplicative sequence with a(p^e) = p if e >= p, otherwise 1.

1, -1, -1, -1, -1, 1, -1, 1, 0, 1, -1, 1, -1, 1, 1, 1, -1, 0, -1, 1, 1, 1, -1, -1, 0, 1, -2, 1, -1, -1, -1, -1, 1, 1, 1, 0, -1, 1, 1, -1, -1, -1, -1, 1, 0, 1, -1, -1, 0, 0, 1, 1, -1, 2, 1, -1, 1, 1, -1, -1, -1, 1, 0, -1, 1, -1, -1, 1, 1, -1, -1, 0, -1, 1, 0, 1, 1, -1, -1, -1, 2, 1, -1, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, 1, 1, 1, -1, 0, 0, 0, -1, -1, -1, -1, -1, 1, -1, 2

Offset: 1

Author

Antti Karttunen, Jan 02 2023

Keywords

• sign
• mult

Comments

Multiplicative because A327936 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A327936.

Cf. A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).

Cf. also A358216, A359433.

Programs

Mathematica

f[p_, e_] := Switch[Mod[e, p], 0, (1 - p)^(e/p), 1, -(1 - p)^((e - 1)/p), , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}] (* Amiram Eldar, Jan 26 2023 *)

PARI

A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };
memoA359432 = Map();
A359432(n) = if(1==n,1,my(v); if(mapisdefined(memoA359432,n,&v), v, v = -sumdiv(n,d,if(dA327936(n/d)*A359432(d),0)); mapput(memoA359432,n,v); (v)));

Formula

Multiplicative with a(p^e) = (1 - p)^(e/p) if p | e, -(1 - p)^((e - 1)/p) if e == 1 (mod p), and 0 otherwise. - Amiram Eldar, Jan 26 2023