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%I #40 Jan 11 2023 06:40:44
%S 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,
%T 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,
%U 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
%N 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).
%C Completely additive modulo 3.
%C The equivalent sequence modulo 2 is A096268 (with offset 1), which produces the {A003159, A036554} classification.
%C 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.
%C 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.
%C 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.
%C Further properties are given in the sequences that list the classes: A332820, A332821, A332822.
%C The scaled imaginary part of the Eisenstein integer-valued function, f, defined in A353445. - _Peter Munn_, Apr 27 2022
%H Antti Karttunen, <a href="/A332823/b332823.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = A102283(A048675(n)) = -1 + (1 + A048675(n)) mod 3.
%F a(1) = 0; for n > 1, a(n) = A102283[(A067029(n) * (2-(A000035(A055396(n))))) + a(A028234(n))].
%F For all n >= 1, k >= 1: (Start)
%F a(n * k) == a(n) + a(k) (mod 3).
%F a(A331590(n,k)) == a(n) + a(k) (mod 3).
%F a(n^2) = -a(n).
%F a(A003961(n)) = -a(n).
%F a(A297845(n,k)) = a(n) * a(k).
%F (End)
%F For all n >= 1: (Start)
%F a(A000040(n)) = (-1)^(n-1).
%F a(A225546(n)) = a(n).
%F a(A097248(n)) = a(n).
%F a(A332461(n)) = a(A332462(n)) = A332814(n).
%F (End)
%F a(n) = A332814(A332462(n)). [Compare to the formula above. For a proof, see A353350.] - _Antti Karttunen_, Apr 16 2022
%o (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); };
%Y Cf. A332813 (0,1,2 version of this sequence), A353350.
%Y Cf. A353354 (inverse Möbius transform, gives another 3-way classification indicator function).
%Y Cf. A102283, A048675.
%Y 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.
%Y Comparable functions: A008836, A064179, A096268, A332814.
%Y A000035, A003961, A028234, A055396, A067029, A097248, A225546, A297845, A331590 are used to express relationship between terms of this sequence.
%Y The formula section also details how the sequence maps the terms of A000040, A332461, A332462.
%K sign
%O 1,1
%A _Antti Karttunen_ and _Peter Munn_, Feb 25 2020
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