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, 310-332.
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
KEYWORD
base,nonn
AUTHOR
Jonathan Vos Post, Aug 16 2005
EXTENSIONS
Corrected and extended by Robert G. Wilson v, Jan 25 2006
STATUS
approved