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A339790 First coefficient of the lindep transform of sigma(n). 3


%S 1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,3,1,2,1,2,2,1,1,2,1,1,2,2,

%T 3,2,1,2,2,4,1,3,1,1,4,2,1,2,1,1,5,1,1,4,3,1,5,2,1,1,1,2,3,1,3,1,1,1,

%U 5,1,1,3,1,2,3,1,4,1,1,3,2,2,1,3,4,2,5,1,1,5,4,6,3,2,4,3,1,4,7,6

%N First 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 as:

%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 "lindep transform".

%H Benoit Cloitre, <a href="/A339790/a339790.png">a(n)/sqrt(n) every 1000 up to 6.10^6</a>

%F Conjecture: a(n) << sqrt(n) with 0 < limsup n-->infty a(n)/sqrt(n) < infty exists (see graphic). Trivially liminf 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]))[1]

%Y Cf. A000203, A339791, A339792.

%K nonn

%O 1,9

%A _Benoit Cloitre_, Dec 17 2020

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