A093017 Luhn algorithm double-and-add sum of digits of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 7, 8, 9, 10, 11, 12, 13

Offset

0,3

Comments

Starting on the right, sum digits after doubling alternating digits beginning with the second. If doubled digit >9, reduce by 9 (sum of digits).

a(n) = A007953(A249873(n)); A093019(n) = 10 - a(10*n) mod 10 if less than 10, otherwise 0. - Reinhard Zumkeller, Nov 08 2014

First differences are b(n) defined for n>0 as follows. Take the prime factorization of n and let x be the number of 2's, y be the number of 5's, and z be min(x,y). If z is even, b(n) = 1 - 9*z. If z is odd and y=z, b(n) = 2 - 9*z. If z is odd and y>z, b(n) = -7 - 9*z. Now a(n) = a(n-1) + b(n). - Mathew Englander, Aug 04 2021

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

John Kilgo, Using the Luhn Algorithm, DotNetJohn.com.

Webopedia, Luhn formula

Wikipedia, Luhn algorithm

Index entries for sequences related to decimal expansion of n

Formula

a(0)=0; for n not divisible by 10, a(n)=1+a(n-1); for n divisible by 10 but not 50, a(n)=2+a(n-10); for n divisible by 50 but not 100, a(n)=1+a(n-50); for n divisible by 100, a(n)=a(n/100). - Mathew Englander, Aug 04 2021

Example

a(18) = 2*1 + 8 = 10.
a(59) = (1+0) + 9 = 10 (1 and 0 are the digits in 10 = 2*5).

Prog

(Haskell)
a093017 n = if n == 0 then 0 else a093017 n' + a007953 (2 * t) + d
            where (n', td) = divMod n 100; (t, d) = divMod td 10
-- Reinhard Zumkeller, Nov 08 2014

Crossrefs

Cf. A093018-A093029.

Cf. A007953, A093019.

Sequence in context: A189506 A173529 A273005 * A156230 A028897 A244158

Adjacent sequences: A093014 A093015 A093016 * A093018 A093019 A093020

Keyword

easy,nonn,base

Author

Ray Chandler, Apr 03 2004

Status

approved