OFFSET
1,4
COMMENTS
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:
(i) x(n) >= 1;
(ii) x(n) + abs(y(n)) + abs(z(n)) is minimal;
(iii) x(n)*b(n) + y(n)*n + z(n) = 0;
(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).
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".
LINKS
Benoit Cloitre, a(n)/sqrt(n) every 1000 up to 6*10^6.
FORMULA
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.
PROG
(PARI) a(n)=(lindep([sigma(n), n, 1])*sign(lindep([sigma(n), n, 1])[1]))[2]
CROSSREFS
KEYWORD
sign
AUTHOR
Benoit Cloitre, Dec 17 2020
STATUS
approved