

A109809


Primes at Levenshtein distance n from previous value when considered as a decimal string.


4



2, 3, 11, 223, 1009, 22111, 100003, 2211127, 10000019, 221111257, 1000000009, 22111111123, 100000000019, 2211111111227, 10000000000051, 221111111111197, 1000000000000223, 22111111111111117, 100000000000000003, 2211111111111111211, 10000000000000000087, 221111111111111111249
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OFFSET

1,1


COMMENTS

For positive n, the string length of a(n+1) is always the 1 + the string length of a(n). This sequence is infinite.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Michael Gilleland, Levenshtein Distance, in Three Flavors. [It has been suggested that this algorithm gives incorrect results sometimes.  N. J. A. Sloane]
V. I. Levenshtein, Efficient reconstruction of sequences from their subsequences or supersequences, J. Combin. Theory Ser. A 93 (2001), no. 2, 310332.


FORMULA

a(0) = 2, a(n+1) = least prime p such that LD(a(n), p) = n, where LD(A,B) = Levenshtein distance from A to B as decimal strings.


EXAMPLE

a(1) = 3 because we transform a(0) = 2 to 3 (a prime) with one substitution.
a(2) = 11 because we transform a(1) = 3 to the least prime 11 with 1 substitution plus one insertion.
a(3) = 223 because we transform a(2) = 11 to the prime 223 with 2 substitutions plus one insertion and any smaller prime can be transformed from 11 in fewer than 3 steps.


MATHEMATICA

levenshtein[s_List, t_List] := Module[{d, n = Length@s, m = Length@t}, Which[s === t, 0, n == 0, m, m == 0, n, s != t, d = Table[0, {m + 1}, {n + 1}]; d[[1, Range[n + 1]]] = Range[0, n]; d[[Range[m + 1], 1]] = Range[0, m]; Do[ d[[j + 1, i + 1]] = Min[d[[j, i + 1]] + 1, d[[j + 1, i]] + 1, d[[j, i]] + If[ s[[i]] === t[[j]], 0, 1]], {j, m}, {i, n}]; d[[ 1, 1]] ]];
NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ@k, k++ ]; k]; a[0] = 2; a[n_] := a[n] = Block[{q = IntegerDigits[a[n  1]][[1]], id = IntegerDigits@a[n  1]}, p = NextPrim[ If[q == 1, Floor[199*10^(n  1)/90  1], 10^(n  1)]]; While[ levenshtein[id, IntegerDigits@p] != n, p = NextPrim@p]; p]; Table[ a[n], {n, 0, 19}] (* Robert G. Wilson v, Jan 25 2006 *)


CROSSREFS

Cf. A000040, A081355, A081356, A081230.
Sequence in context: A066100 A029497 A318130 * A211349 A096456 A262201
Adjacent sequences: A109806 A109807 A109808 * A109810 A109811 A109812


KEYWORD

base,nonn


AUTHOR

Jonathan Vos Post, Aug 16 2005


EXTENSIONS

Corrected and extended by Robert G. Wilson v, Jan 25 2006


STATUS

approved



