A353354 Inverse Möbius transform of A332823.

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

Offset

1

Comments

a(n) is determined by the cubefree part of n, and has the range {-1, 0, 1}.

Proof: A332823 is the scaled imaginary part of a completely multiplicative function, f, from the positive integers to the Eisenstein integers (the range of f being the cube roots of unity). Let g be the inverse Moebius transform of f, which is therefore multiplicative. As a function, "scaling the imaginary part" is a homomorphism with respect to addition, so (a(n)) -- being the inverse Moebius transform of A332823 -- is a scaled imaginary part of g. We can show the range of g is the 7 Eisenstein integers closest to 0, namely the 6 sixth roots of unity and 0 itself. We deduce (a(n)) has the range {-1, 0, 1} (in contrast to say, A353364).

See A353446, which is twice the real part of g, for further details.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Eric Weisstein's World of Mathematics, Eisenstein Integer

Wikipedia, Eisenstein integer

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = Sum_{d|n} A332823(d).

a(n) = A353328(n) - A353329(n) = A353328(n) - A353328(A003961(n)).

a(n) = A008966(m) * A128834(A090882(m)) = A008966(m) * A128834(A195017(m) mod 6), where m = A050985(n), the cubefree part of n, and A008966(.) is the characteristic function of squarefree numbers.

For all n >= 1, a(A003961(n)) = -a(n); and for all m >= 1, a(n*m^3) = a(n).

Prog

(PARI)
A332823(n) = { my(f = factor(n), u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u, -1, u); };
A353354(n) = sumdiv(n, d, A332823(d));

Crossrefs

Sequences used in a formula defining this sequence: A008966, A050985, A090882, A128834, A195017, A332823, A353328, A353329.

Cf. A003961, A048675, A353352, A353446.

Positions of particular values: A353355 (0), A353356 (1), A353357 (-1).

Somewhat analogous sequence: A353364.

Sequence in context: A343173 A285255 A324823 * A284912 A192082 A353498

Adjacent sequences: A353351 A353352 A353353 * A353355 A353356 A353357

Keyword

sign

Author

Antti Karttunen and Peter Munn, Apr 15 2022

Status

approved