A262083 Smallest possible prime factor of 10^k+n for any k.

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, 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, 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

Offset: 0

Author

Sergio Pimentel, Sep 10 2015

Keywords

• nonn

Comments

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

From Robert G. Wilson v, Sep 13 2015: (Start)

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

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

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

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

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),

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

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),

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),

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),

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

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

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

(End)

For complete lists of when a(n) < 19, see Wilson's Congruencies a-file. - Danny Rorabaugh, Oct 08 2015

Examples

			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).
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..10000
• Robert G. Wilson v, Congruencies for A262083

Crossrefs

Cf. A000533, A003617, A038371, A185121.

Programs

Mathematica

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 *)

Extensions

More terms from Robert G. Wilson v, Sep 13 2015