A349912 Sum of A336466 and its Dirichlet inverse, where A336466 is fully multiplicative with a(p) = oddpart(p-1).

2, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 0, 6, 2, 1, 0, 1, 0, 1, 6, 10, 0, 1, 1, 6, 1, 3, 0, 0, 0, 1, 10, 2, 6, 1, 0, 18, 6, 1, 0, 0, 0, 5, 1, 22, 0, 1, 9, 1, 2, 3, 0, 1, 10, 3, 18, 14, 0, 1, 0, 30, 3, 1, 6, 0, 0, 1, 22, 0, 0, 1, 0, 18, 1, 9, 30, 0, 0, 1, 1, 10, 0, 3, 2, 42, 14, 5, 0, 1, 18, 11, 30, 46, 18, 1, 0, 9, 5, 1

Offset: 1

Author

Antti Karttunen, Dec 08 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A336466 (also a quadrisection of this sequence), A349911.

Cf. also A322581.

Programs

Mathematica

f[p_, e_] := ((p-1)/2^IntegerExponent[p-1, 2])^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

PARI

A000265(n) = (n>>valuation(n,2));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,A000265(f[k,1]-1)^f[k,2]); };
memoA349911 = Map();
A349911(n) = if(1==n,1,my(v); if(mapisdefined(memoA349911,n,&v), v, v = -sumdiv(n,d,if(dA336466(n/d)*A349911(d),0)); mapput(memoA349911,n,v); (v)));
A349912(n) = (A336466(n)+A349911(n));

Formula

a(n) = A336466(n) + A349911(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A336466(d) * A349911(n/d).

a(4*n) = A336466(n).