

A332823


A 3way 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).


8



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
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OFFSET

1,3


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 wellstudied 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 FermiDirac factorization of a member of one class gives a member of the other class. So, if 4 is not a FermiDirac 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 FermiDirac 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.


LINKS

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


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)^(n1).
a(A225546(n)) = a(n).
a(A097248(n)) = a(n).
a(A332461(n)) = a(A332462(n)) = A332814(n).
(End)


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); };


CROSSREFS

Cf. A332813 (0,1,2 version of this sequence).
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.
Sequence in context: A140074 A342004 A284881 * A090174 A165556 A127243
Adjacent sequences: A332820 A332821 A332822 * A332824 A332825 A332826


KEYWORD

sign


AUTHOR

Antti Karttunen and Peter Munn, Feb 25 2020


STATUS

approved



