A353446 Let g be the inverse Möbius transform of the Eisenstein integer-valued function f defined in A353445. a(n) is twice the real part of g(n).

2, 1, 1, 0, 1, 2, 1, 2, 0, -1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 0, -1, -1, 1, 1, 0, 2, 2, 0, 1, 1, 1, 0, 2, -1, 2, 0, 1, 2, -1, 1, 1, 1, 1, 0, 0, -1, 1, 2, 0, 0, 2, 0, 1, 1, -1, 1, -1, 2, 1, 0, 1, -1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 1, 2, 0, 0, 2, 1, 1, -1, 1, -1, 1, 0, -1, 2, -1, 1, 1, 0, -1, 0, 2, -1, 2, 0, 1, 0, 0, 0, 1

Offset: 1

Author

Antti Karttunen and Peter Munn, Apr 19 2022

Keywords

• sign

Comments

The imaginary part of g(n) is A353354(n)*(sqrt(3)/2)*i.

f(n), g(n), and so also a(n), are determined by the cubefree part of n, A050985(n). If the cubefree part is not squarefree, g(n) is 0; otherwise g(n) = x^(A195017(A050985(n))), where x = (1 + sqrt(3)*i)/2, the primitive 6th root of unity with positive imaginary part.

The above formula arises from g being multiplicative (because f is multiplicative). g(prime(m)^k) is 1 for k == 0 (mod 3), 0 for k == 2 (mod 3), and for k == 1 (mod 3) the result depends on the parity of m. g(prime(m)^(3k+1)) is 1+w for odd m, -w for even m, where w is the cube root of unity with positive imaginary part. 1+w and -w are the primitive 6th roots of unity.

So the range of g is the 6 sixth roots of unity and 0 itself: these are the 7 Eisenstein integers closest to 0, and they are clearly closed under multiplication. The range of (a(n)) is [-2..2]. g(n) and a(n) are 0 if and only if the cubefree part of n is not squarefree. (Compare with the Moebius function being 0 when its argument is not squarefree.) Otherwise a(n) is even if and only if n is in A332820.

Links

• Peter Munn, Table of n, a(n) for n = 1..10000
• Peter Munn, Figure showing relationship of the Eisenstein integer g(n) to the presence of n in other sequences.
• Eric Weisstein's World of Mathematics, Eisenstein Integer

Crossrefs

Sequences used in a definition of this sequence: A008966, A050985, A090882, A087204, A195017, A353445.

See A353354 for the imaginary part.

Positions of particular values depend on A059269, A211337, A211338, A332820 as shown in the formula section.

Positions of even numbers: A353355.

Programs

PARI

A050985(n) = { my(f=factor(n)); f[, 2] = apply(x->(x % 3), f[, 2]); factorback(f); }; \\ From A050985
A087204(n) = ([2, 1, -1, -2, -1, 1][1+(n%6)]);
A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); };
A353446(n) = { my(u=A050985(n)); issquarefree(u) * A087204(abs(A195017(u))); };

Formula

a(n) = A008966(m) * A087204(A090882(m)) = A008966(m) * A087204(|A195017(m)|), where m = A050985(n), the cubefree part of n, and A008966(.) is the characteristic function of squarefree numbers.

For n >= 1, -2 <= a(n) <= 2.

{n : a(n) = -2} = {A211338} INTERSECT {A332820}.

{n : a(n) = -1} = {A211337} \ {A332820}.

{n : a(n) = 0} = {A059269}.

{n : a(n) = 1} = {A211338} \ {A332820}.

{n : a(n) = 2} = {A211337} INTERSECT {A332820}.