A262083 - OEIS

%I #25 Oct 31 2015 15:17:13

%S 2,7,2,7,2,3,2,17,2,7,2,3,2,7,2,5,2,3,2,7,2,11,2,3,2,5,2,7,2,3,2,7,2,

%T 7,2,3,2,7,2,7,2,3,2,7,2,5,2,3,2,13,2,7,2,3,2,5,2,7,2,3,2,7,2,17,2,3,

%U 2,7,2,7,2,3,2,7,2,5,2,3,2,7,2,7,2,3,2,5,2,7,2,3,2,17,2,7,2,3,2,7,2,7,2

%N Smallest possible prime factor of 10^k+n for any k.

%C Is this sequence bounded?  What are the records for a(n)?

%C From _Robert G. Wilson v_, Sep 13 2015: (Start)

%C First occurrence of the i-th prime: 0, 5, 15, 1, 21, 49, 7, 357, 24871, 364021, ..., .

%C a(n) = 2 when n == 0 (mod 2),

%C a(n) = 3 when n == 5 (mod 6),

%C a(n) = 5 when n == 15 or 25 (mod 30),

%C a(n) = 7 when n == 1, 3, 9, 13, 19, 27, 31, 33, 37, 39, 43, 51, 57, 61, 67, 69, 73, 79, 81, 87, 93, 97, 99, 103, 109, 111, 117, 121, 123, 127, 129, 139, 141, 151, 153, 157, 159, 163, 169, 171, 177, 181, 183, 187, 193, 199, 201 or 207 (mod 210),

%C a(n) = 11 when n = 21, 133, 441, 483, 637, 903, 1057, 1099, 1407, 1519, 1561, 1827, 1869, 1981, 2023 or 2289 (mod 2310),

%C a(n) = 13 when n = 49, 147, 217, 231, 259, 399, 469, 511, 651, 679, 693, 763, 777, 861, 987, 1141, 1197, (413 terms missing), 29883 or 29953, ... (mod 30030),

%C a(n) = 17 when n = 7, 63, 91, 189, 273, 301, 343, 427, 553, 567, 609, 721, 819, 847, 889, 931, 973, 1029, (8044 terms missing), 510349 or 510447 (mod 510510),

%C a(n) = 19 when n = 357, 1071, 2737, 3451, 6069, 6307, 8211, 9163, 9639, 10353, 12019, 12733, 13447, 13923, 15351, 15589, 17017, 17493, 18207, ... (mod 9699690),

%C a(n) = 23 when n = 24871, 47481, 74613, 88179, 92701, 106267, 133399, 142443, 160531, 187663, 201229, 210273, 223839, 250971, 264537, 309757, ... (mod 223092870),

%C a(n) = 29 when n = 364021, 988057, ... (mod 6469693230), etc.

%C To the question if this sequence is 'bounded', I would answer no.

%C (End)

%C For complete lists of when a(n) < 19, see Wilson's Congruencies a-file. - _Danny Rorabaugh_, Oct 08 2015

%H Robert G. Wilson v, <a href="/A262083/b262083.txt">Table of n, a(n) for n = 0..10000</a>

%H Robert G. Wilson v, <a href="/A262083/a262083.txt">Congruencies for A262083</a>

%e a(1) = 7 since 10^k+1 is not divisible by 2,3 or 5 for all k but is divisible by 7 when k = 3 (i.e., 1001 = 7*11*13).

%t p = Prime@ Range@ 25; f[n_] := Block[{k = 1, lst = {}}, While[k < 25, AppendTo[lst, Position[ Mod[ PowerMod[10, k, p] + n, p] 0, 1, 1][[1, 1]]]; k++]; lst = Union@ lst; Prime@ lst[[1]]]; Array[f, 101, 0] (* _Robert G. Wilson v_, Sep 13 2015 *)

%Y Cf. A000533, A003617, A038371, A185121.

%K nonn

%O 0,1

%A _Sergio Pimentel_, Sep 10 2015

%E More terms from _Robert G. Wilson v_, Sep 13 2015