A355690 Dirichlet inverse of A152822, where A152822 is the characteristic function of numbers not congruent to 2 mod 4.

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

Offset

1

Comments

Multiplicative because A152822 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A152822(n/d) * a(d).

Multiplicative with a(2^e) = A010892(1+e), and for odd primes p, a(p^e) = -1 if e = 1, otherwise 0. - Antti Karttunen, Dec 23 2022

a(n) = A359605(n) - A359606(n). - Antti Karttunen, Jan 12 2023

Prog

(PARI)
A152822(n) = (2!=(n%4));
memoA355690 = Map();
A355690(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355690, n, &v), v, v = -sumdiv(n, d, if(d<n, A152822(n/d)*A355690(d), 0)); mapput(memoA355690, n, v); (v)));
(PARI)
A010892(n) = ([1, 1, 0, -1, -1, 0][n%6 + 1]);
A355690(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], A010892(1+f[k, 2]), -(1==f[k, 2]))); }; \\ Antti Karttunen, Dec 23 2022

Crossrefs

Cf. A010892, A042965, A152822, A359590 (absolute values), A359605, A359606.

Cf. also A355688, A355689, A355691.

Sequence in context: A070749 A285341 A059778 * A359590 A104521 A317198

Adjacent sequences: A355687 A355688 A355689 * A355691 A355692 A355693

Keyword

sign,mult,easy

Author

Antti Karttunen, Jul 15 2022

Status

approved