﻿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.