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

Author

Ray Chandler, Apr 03 2004

Keywords

• easy
• nonn
• base

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

Examples

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

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

Crossrefs

Cf. A093018-A093029.

Cf. A007953, A093019.

Programs

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

Python

def a(n):
    s = str(n)
    r = s[::-1]
    x = sum(int(d) for d in r[::2])
    x += sum(q if (q:=2*int(d)) < 10 else q-9 for d in r[1::2])
    return x
print([a(n) for n in range(87)]) # Michael S. Branicky, Jul 23 2024

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