A347093 Sum of A322577 (convolution of Dedekind psi with Euler phi) and its Dirichlet inverse.

2, 0, 0, 16, 0, 48, 0, 24, 36, 80, 0, 36, 0, 112, 120, 73, 0, 64, 0, 60, 168, 176, 0, 192, 100, 208, 96, 84, 0, 0, 0, 156, 264, 272, 280, 336, 0, 304, 312, 320, 0, 0, 0, 132, 160, 368, 0, 378, 196, 192, 408, 156, 0, 432, 440, 448, 456, 464, 0, 960, 0, 496, 224, 373, 520, 0, 0, 204, 552, 0, 0, 688, 0, 592, 288, 228, 616

Offset: 1

Author

Antti Karttunen, Aug 18 2021

Keywords

• nonn

Comments

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

Links

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

Crossrefs

Cf. A000010, A001615, A030059, A030229, A322577, A347092, A347094.

Programs

PARI

up_to = 16384;
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA322577(n)));
A347092(n) = v347092[n];
A347093(n) = (A322577(n)+A347092(n));

Formula

a(n) = A322577(n) + A347092(n).

For n > 1, a(n) = -Sum_{d|n, 1A322577(d) * A347092(n/d).

For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A322577(A030229(n)).