A353445 -id:A353445 - cp's OEIS frontend

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

Antti Karttunen and Peter Munn, Apr 19 2022

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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.

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

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.

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))); };

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}.

A332823 A 3-way classification indicator generated by the products of two consecutive primes and the cubes of primes. a(n) is -1, 0, or 1 such that a(n) == A048675(n) (mod 3).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

sign

Completely additive modulo 3.
The equivalent sequence modulo 2 is A096268 (with offset 1), which produces the {A003159, A036554} classification.
Let H be the multiplicative subgroup of the positive rational numbers generated by the products of two consecutive primes and the cubes of primes. a(n) indicates the coset of H containing n. a(n) = 0 if n is in H. a(n) = 1 if n is in 2H. a(n) = -1 if n is in (1/2)H.
The properties of this classification can usefully be compared to two well-studied classifications. With the {A026424, A028260} classes, multiplying a member of one class by a prime gives a member of the other class. With the {A000028, A000379} classes, adding a factor to the Fermi-Dirac factorization of a member of one class gives a member of the other class. So, if 4 is not a Fermi-Dirac factor of k, k and 4k will be in different classes of the {A000028, A000379} set; but k and 4k will be in the same class of the {A026424, A028260} set. For two numbers to necessarily be in different classes when they differ in either of the 2 ways described above, 3 classes are needed.
With the classes defined by this sequence, no two of k, 2k and 4k are in the same class. This is a consequence of the following stronger property: if k is any positive integer and m is a member of A050376 (often called Fermi-Dirac primes), then no two of k, k * m, k * m^2 are in the same class. Also, if p and q are consecutive primes, then k * p and k * q are in different classes.
Further properties are given in the sequences that list the classes: A332820, A332821, A332822.
The scaled imaginary part of the Eisenstein integer-valued function, f, defined in A353445. - Peter Munn, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A332813 (0,1,2 version of this sequence), A353350.
Cf. A353354 (inverse Möbius transform, gives another 3-way classification indicator function).
Cf. A102283, A048675.
Cf. A332820, A332821, A332822 for positions of 0's, 1's and -1's in this sequence; also A003159, A036554 for the modulo 2 equivalents.
Comparable functions: A008836, A064179, A096268, A332814.
A000035, A003961, A028234, A055396, A067029, A097248, A225546, A297845, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A000040, A332461, A332462.

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); };

a(n) = A102283(A048675(n)) = -1 + (1 + A048675(n)) mod 3.
a(1) = 0; for n > 1, a(n) = A102283[(A067029(n) * (2-(A000035(A055396(n))))) + a(A028234(n))].
For all n >= 1, k >= 1: (Start)
a(n * k) == a(n) + a(k) (mod 3).
a(A331590(n,k)) == a(n) + a(k) (mod 3).
a(n^2) = -a(n).
a(A003961(n)) = -a(n).
a(A297845(n,k)) = a(n) * a(k).
(End)
For all n >= 1: (Start)
a(A000040(n)) = (-1)^(n-1).
a(A225546(n)) = a(n).
a(A097248(n)) = a(n).
a(A332461(n)) = a(A332462(n)) = A332814(n).
(End)
a(n) = A332814(A332462(n)). [Compare to the formula above. For a proof, see A353350.] - Antti Karttunen, Apr 16 2022

cp's OEIS Frontend

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).

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