By Derek Orr, Oct 02 2014
The numbers n < 1000 such that a(n) > 1000 or 0 are
{47, 67, 100, 107, 114, 142, 197, 219, 232, 256, 295, 308, 355, 360, 373, 464,
478, 496, 545, 549, 578, 583, 602, 607, 618, 659, 661, 716, 727, 750, 760, 785,
786, 821, 826, 856, 906, 911, 958, 961}.
a(47) > 10000. The number must be of the form (47/99)*(10^(6n+1)-10)+1 for some n.
(47/99)*(10^(2*n+1)-10)+1 is divisible by 3 for n = 3k – 2 (k > 0).
(47/99)*(10^(2*n+1)-10)+1 is divisible by 37 for n = 3k – 1 (k > 0).
a(67) > 10000. The number must be of the form (67/99)*(10^(2*n+1)-10)+1 for n
not congruent to 2 mod 3. If n == 2 (mod 3), the term is divisible by 3.
a(100) > 10000. The number must be of the form (100/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 7.
If n is congruent to 2 mod 6, the term is divisible by 3.
a(107) = 2478.
a(114) = 1164.
a(142) > 10000. The number must be of the form (142/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 7. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(197) > 10000. The number must be of the form (197/999)*(10^(3*n+1)-10)+1 for n
not congruent to 1 mod 3. If n == 1 (mod 3), the term is divisible by 3.
a(219) > 10000. The number must be of the form (219/999)*(10^(6*n+1)-10)+1 for some n.
(219/999)*(10^(3*n+1)-10)+1 is divisible by 7 for n = 2k+1.
a(232) > 10000. The number must be of the form (232/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 11. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(256) = 1260.
a(295) = 1924.
a(308) > 10000. The number must be of the form (308/999)*(10^(3*n+1)-10)+1 for n
congruent to 2 mod 6 or 0 mod 6. If n is odd, the term is divisible by 13. If n is
congruent to 4 mod 6, the term is divisible by 3.
a(355) > 10000. The number must be of the form (355/999)*(10^(3*n+1)-10)+1 for n
not congruent to 2 mod 3. If n == 2 (mod 3), the term is divisible by 3.
a(360) > 10000. The number must be of the form (360/999)*(10^(6*n+1)-10)+1 for some n.
(360/999)*(10^(3*n+1)-10)+1 is divisible by 13 for n = 2k+1.
a(373) > 10000. The number must be of the form (373/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 7. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(464) = 1136.
a(478) > 10000. The number must be of the form (478/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 7. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(496) > 10000. The number must be of the form (496/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 11. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(545) > 10000. The number must be of the form (545/999)*(10^(3*n+1)-10)+1 for n
not congruent to 1 mod 3. If n == 1 (mod 3), the term is divisible by 3.
a(549) > 10000. The number must be of the form (549/999)*(10^(3*n+1)-10)+1 for some n.
a(578) = 2285.
a(583) = 5874.
a(602) = 1115.
a(607) > 10000. The number must be of the form (607/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 13. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(618) = 4212.
a(659) = 1782.
a(661) = 7468.
a(716) > 10000. The number must be of the form (716/999)*(10^(18*n+1)-10)+1 for some n.
(716/999)*(10^(3*n+1)-10)+1 is divisible by 7 if n is odd, 19 if n == 2 (mod 6), and 3 if n == 4 (mod 6).
a(727) > 10000. The number must be of the form (727/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 11. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(750) = 5156.
a(760) > 10000. The number must be of the form (760/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 11. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(785) = 1064.
a(786) = 2412.
a(821) = 5174.
a(826) > 10000. The number must be of the form (826/999)*(10^(18*n+1)-10)+1 for some n.
(826/999)*(10^(3*n+1)-10)+1 is divisible by 11 if n is odd, 3 if n == 2 (mod 6), and 19 if n == 4 (mod 6).
a(856) = 1894.
a(906) > 10000. The number must be of the form (906/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 13. If n is
congruent to 2 mod 6, the term is divisible by 19.
a(911) > 10000. The number must be of the form (911/999)*(10^(3*n+1)-10)+1 for n not
congruent to 1 mod 3. If n == 1 (mod 3), the term is divisible by 3.
a(958) > 10000. The number must be of the form (958/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 11. If n is
congruent to 2 mod 6, the term is divisible by 3.
a(961) > 10000. The number must be of the form (961/999)*(10^(3*n+1)-10)+1 for n
congruent to 4 mod 6 or 0 mod 6. If n is odd, the term is divisible by 7. If n is
congruent to 2 mod 6, the term is divisible by 3.