A226241 Primes that cannot be reached from 2 via a chain of primes obtained adding or deleting a digit from the end or the beginning of the previous term of the chain.

89, 101, 103, 107, 109, 151, 163, 227, 251, 257, 263, 269, 281, 307, 389, 401, 409, 457, 503, 509, 521, 557, 563, 569, 587, 601, 607, 701, 709, 809, 821, 827, 857, 863, 881, 887, 907, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069

Offset: 1

Author

Giovanni Resta, Jun 01 2013

Keywords

• nonn
• base

Comments

All the primes containing a 0 are members since a 0 cannot be added at the end (it is even) nor at the beginning since, for example, 02 and 0013 are not canonical representations. The sequence is infinite, since there are exactly 820293 other primes that can be reached from 2, the largest one being 5481899436575987524681453773937333.

Examples

			All the primes < 89 can be reached from 2. For example, 2 -> 23 -> 3 -> 37.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000
• Carlos Rivera, Puzzle 690 - Unreachable Primes

Crossrefs

Cf. A226144, A050249.

Programs

Mathematica

step[p_] := Block[{dn = 10^IntegerLength@p}, Select[ Union[{Floor[p/10], Mod[p, dn/10]}, p*10 + {1, 3, 7, 9}, Range[9]*dn + p], PrimeQ[#] &]]; old = {}; new = {2}; wrk = {}; While[new != {}, wrk = Flatten[step /@ new]; old = Union[new, old]; new = Complement[wrk, old]; Print["# = ", Length@old, "  max = ", Max[old], " new # = ", Length@new]]; Print["Missing up to 1000 = ", Complement[Prime@Range[168], old]]