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

1,5

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.

Wikipedia, Dirichlet convolution.

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A083254(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)

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A083254(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(d<n, A083254(n/d)*A323912(d), 0)); mapput(memoA323912, n, v); (v))); \\ Antti Karttunen, Nov 22 2024

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.

Sequence in context: A318304 A318989 A082493 * A021887 A085015 A378644

Adjacent sequences: A323909 A323910 A323911 * A323913 A323914 A323915

Keyword

sign

Author

Antti Karttunen, Feb 12 2019

Status

approved