OFFSET
1,1
COMMENTS
Numbers N such that there exists a residue r and a modulus s with s^3 > N > s > r > 0 and gcd(r,s)=1 such that N has at least 6 divisors in the residue class r modulo s.
It can be shown that this sequence is infinite.
No case is known with more than 6 such divisors.
The first 10 terms were found by Hendrik Lenstra.
Additional terms up to a(20) were supplied by David Broadhurst.
LINKS
Henri Cohen, Diviseurs appartenant à une même classe résiduelle, Séminaire de Théorie des Nombres de Bordeaux (1982-1983), Volume 12, pp. 1-12.
D. Coppersmith, N. Howgrave-Graham and S. V. Nagaraj, Divisors in residue classes, constructively, Math. Comp., 77 (2008), 531-545.
H. W. Lenstra, Divisors in residue classes, Math. Comp., 42 (1984), 331-340. See Table 2.
Paul Zimmermann, divisorslenstra.c.
EXAMPLE
a(7) = 1796760 has 6 divisors congruent to 3 modulo 137, namely 3, 140, 414, 3565, 7812, 19320, and 6 divisors congruent to 93 modulo 137, namely 93, 230, 504, 4340, 12834, 598920.
a(17) = 7022400 has 6 divisors for 4 classes r mod s=199, namely r=4, 8, 11 and 22. - Paul Zimmermann, Jan 18 2018
PROG
(C) See the Paul Zimmermann link.
CROSSREFS
KEYWORD
hard,nice,nonn
AUTHOR
David Broadhurst, Oct 31 2008
EXTENSIONS
Previous values checked by and a(21) from Paul Zimmermann, Jan 18 2018
STATUS
approved