A231963 - OEIS

%I #20 Oct 03 2023 08:49:24

%S 17,24,31,48,55,62,79,86,93,109,116,123,130,147,154,161,178,185,192,

%T 208,215,222,239,246,253,260,277,284,291,307,314,321,338,345,352,369,

%U 376,383,390,406,413,420,437,444,451,468,475,482,499,505,512,529,536,543,550,567,574,581

%N Concatenate n with its UPC check digit, a(n) = 10*n + A237042(n).

%C Theoretically, a UPC check digit can be used with 1- or 2-digit numbers as well as numbers with more than 17 digits. There is probably no practical need for the former, while the latter would probably require a more robust error-detecting (if not error-correcting) mechanism.

%C However, the UPC check digit is a more robust mechanism than a simple digital root would be, as it guards against dyslexia when a seemingly scannable UPC code does not scan and the human operator has to type in the code (whether a cash register or a mobile scanner).

%C This is because to generate the check digit, the digits in the one's place, hundred's place, ten hundred's place, etc., are multiplied by 3. Thus, for example, 13 gives 0 for a check digit while 31 gives 4 for a check digit.

%C Some manufacturers that have or might have UPCs ending in the numbers shown above include H. J. Heinz (013000) and the Hershey Company (068000).

%C As a rule of thumb, the terms mostly advance in steps of 7, sometimes 17.

%D David Salomon, Coding for Data and Computer Communications. New York: Springer (2006): 41 - 42.

%H GS1, <a href="http://www.gs1.org/barcodes/support/check_digit_calculator">Check Digit Calculator</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UPC.html">UPC</a>.

%F a(n) = 10n + c(n), where c(n) = A237042(n) = -( (Sum_{i=1..floor(L/2)} d(2i-1)) + 3*(Sum_{j=0..floor(L/2)} d(2j)) ) mod 10, where L is how many digits n has, d(L - 1) is the most significant digit of n, ..., and d(0) is the one's place digit.

%e a(13) = 130 because 1 * 1 + 3 * 3 = 10, giving a check digit of 0.

%e a(14) = 147 because 1 * 1 + 4 * 3 = 13, and -13 = 7 mod 10.

%e a(15) = 154 because 1 * 1 + 5 * 3 = 16, and -16 = 4 mod 10.

%o (PARI) a(n) = 10*n + vecsum(digits(n,100)*31\-10) % 10; \\ _Kevin Ryde_, Oct 03 2023

%Y Cf. A237042, A093018, A108773.

%K nonn,base,easy

%O 1,1

%A _Alonso del Arte_, Nov 15 2013