A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

1, 0, -1, 0, -3, 2, -5, 0, -2, 2, -9, 4, -11, 2, 5, 0, -15, 2, -17, 4, 7, 2, -21, 8, -6, 2, -4, 4, -27, -2, -29, 0, 11, 2, 17, 8, -35, 2, 13, 8, -39, -6, -41, 4, 8, 2, -45, 16, -10, -2, 17, 4, -51, 0, 29, 8, 19, 2, -57, 4, -59, 2, 12, 0, 35, -14, -65, 4, 23, -10, -69, 24, -71, 2, 4, 4, 47, -18, -77, 16, -8, 2, -81, -4, 47, 2, 29, 8, -87, 4

Offset: 1

Author

Antti Karttunen, Feb 12 2019

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.
• Wikipedia, Dirichlet convolution.

Crossrefs

Cf. A083254, A323913.

Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.

Programs

PARI

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];

PARI

A083254(n) = (2*eulerphi(n)-n);
memoA323912 = Map();
A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(dA083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083254(n/d) * a(d).

From Antti Karttunen, Nov 22 2024: (Start)

Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:

a(n) = Sum_{d|n} A323910(d).

a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).

a(n) = Sum_{d|n} A055615(d)*A067824(n/d)

a(n) = Sum_{d|n} A046692(d)*A253249(n/d)

a(n) = Sum_{d|n} A130054(d)*A174725(n/d)

a(n) = Sum_{d|n} A101035(d)*A330575(n/d)

a(n) = Sum_{d|n} A191161(d)*A328722(n/d)

(End)