A160094 a(n) = 1 + A122840(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

Offset

1,10

Comments

a(n) is the Levenshtein distance from the decimal expansion of n - 1 to the decimal expansion of n. For example, to convert "9" to "10", substitute "0" for "9" and insert "1". Since two such operations are required, a(10) = 2. See the analogous A091090 (binary expansion) and A115777 (full definition). - Rick L. Shepherd, Mar 25 2015

Links

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Formula

From Hieronymus Fischer, Jun 08 2012: (Start)

With m = floor(log_10(n)), frac(x) = x-floor(x):

a(n) = Sum_{j=0..m} (1 - ceiling(frac(n/10^j))).

a(n) = m + 1 + Sum_{j=1..m} (floor(-frac(n/10^j))).

a(n) = 1 + A054899(n) - A054899(n-1).

G.f.: g(x) = (x/(1-x)) + Sum_{j>0} x^10^j/(1-x^10^j). (End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 10/9. - Amiram Eldar, Jul 10 2023

Example

a(160) = 2 because the last nonzero digit of 160 (counting from left to right), when 160 is written in base 10, is 6, and that 6 occurs 2 digits from the right in 160.

Mathematica

IntegerExponent[Range[150]]+1 (* Harvey P. Dale, Feb 06 2015 *)

Prog

(Notepad++)
For(n := 1, n < 10001, Inc(n), Echo(n +> ' ' +> Levenshtein(n-1, n)))
Copy the above line into an editing buffer of Notepad++ with the NppCalc plugin installed and ActiveCalc enabled. Position the cursor at the end of the line and press enter to duplicate the contents of this b-file. - Rick L. Shepherd, Mar 25 2015

Crossrefs

Cf. A054899, A055640, A055641, A102669, A122840, A122841, A160093, A196563, A196564, A091090, A115777.

Sequence in context: A113607 A351352 A082586 * A043283 A127937 A250209

Adjacent sequences: A160091 A160092 A160093 * A160095 A160096 A160097

Keyword

base,easy,nonn,changed

Author

Anonymous, May 01 2009

Extensions

Name simplified by Jon E. Schoenfield, Feb 26 2014

Status

approved