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Second coefficient of the lindep transform of sigma(n).
3

%I #17 Apr 27 2023 07:05:45

%S -1,-1,-1,-2,-1,-2,-1,-2,-3,-2,-1,-2,-1,-2,-3,-2,-1,-2,-1,-2,-3,-5,-1,

%T -5,-1,-3,-3,-2,-1,-5,-1,-2,-3,-3,-4,-5,-1,-3,-3,-9,-1,-7,-1,-2,-7,-3,

%U -1,-5,-1,-2,-7,-2,-1,-9,-4,-2,-7,-3,-1,-3,-1,-3,-5,-2,-4,-2,-1,-2,-7,-2,-1,-8,-1,-3,-5,-2,-5,-2,-1,-7

%N Second coefficient of the lindep transform of sigma(n).

%C If b(n) is a sequence of integers, we will call the "lindep transform" of b(n) the triplet of sequences (x(n), y(n), z(n)) such that:

%C (i) x(n) >= 1;

%C (ii) x(n) + abs(y(n)) + abs(z(n)) is minimal;

%C (iii) x(n)*b(n) + y(n)*n + z(n) = 0;

%C (iv) if with the conditions (i), (ii), (iii) there exist several triplets (x(n), y(n), z(n)) we then choose the one with minimal y(n).

%C We call x(n) the first coefficient of the lindep transform of b(n), y(n) the second and z(n) the third. As this corresponds to the lindep function of PARI/GP this transform is called the "lindep transform".

%H Benoit Cloitre, <a href="/A339791/a339791.png">a(n)/sqrt(n) every 1000 up to 6*10^6</a>.

%F Conjecture: a(n) << sqrt(n) with -oo < liminf n->oo a(n)/sqrt(n) < 0 exists (see graphic). Trivially limsup a(n)/sqrt(n) = 0 since for prime n we have a(n)=-1.

%o (PARI) a(n)=(lindep([sigma(n), n, 1])*sign(lindep([sigma(n), n, 1])[1]))[2]

%Y Cf. A000203, A339790, A339792.

%K sign

%O 1,4

%A _Benoit Cloitre_, Dec 17 2020