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).

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

Author

Antti Karttunen and Peter Munn, Feb 25 2020

Keywords

• sign

Comments

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

Links

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

Crossrefs

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.

Programs

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); };

Formula

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