A108773 -id:A108773 - cp's OEIS frontend

A136614 Sum of digits of A108773 and of A136613.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 2, 4, 6, 8, 10, 12, 14, 16, 18, 11, 4, 6, 8, 10, 12, 14, 16, 18, 11, 13, 6, 8, 10, 12, 14, 16, 18, 11, 13, 15, 8, 10, 12, 14, 16, 18, 11, 13, 15, 17, 10, 12, 14, 16, 18, 11, 13, 15, 17, 19, 12, 14, 16, 18, 11, 13, 15, 17, 19, 21, 14, 16, 18, 11

Offset: 0

Reinhard Zumkeller, Jan 13 2008

a(n) = A007953(A108773(n)) = A007953(A136613(n)).

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

A108203 Numbers n put into lexicographical order which are the concatenation of k and the sum of the digits of k.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 112, 123, 134, 145, 156, 167, 178, 189, 202, 213, 224, 235, 246, 257, 268, 279, 303, 314, 325, 336, 347, 358, 369, 404, 415, 426, 437, 448, 459, 505, 516, 527, 538, 549, 606, 617, 628, 639, 707, 718, 729, 808, 819, 909, 1001

Offset: 1

Eric Angelini and Robert G. Wilson v, Jun 15 2005

			1 --> 11, 2 --> 22, 3 --> 33, ..., 9 --> 99, 10 --> 101, 11 --> 112, 12 --> 123,
..., 18 --> 189, 19 --> 1910 (ouch!), 20 --> 202, 21 --> 213, ...

Cf. A062028, A064806. Equals A108773 sorted.

f[n_] := FromDigits[ Join[ IntegerDigits[ n], IntegerDigits[Plus @@ IntegerDigits[ n]] ]]; t = {}; Do[t = Union[AppendTo[t, f[n]]], {n, 10^6}]

a(55) from Rémy Sigrist, May 16 2019

A136613 Concatenation of (sum of digits of n) and n.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 211, 312, 413, 514, 615, 716, 817, 918, 1019, 220, 321, 422, 523, 624, 725, 826, 927, 1028, 1129, 330, 431, 532, 633, 734, 835, 936, 1037, 1138, 1239, 440, 541, 642, 743, 844, 945, 1046, 1147, 1248, 1349, 550, 651, 752

Offset: 0

Reinhard Zumkeller, Jan 13 2008

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

Cf. A007953, A108773, A136614.

a[n_]:=FromDigits[Join[{DigitSum[n]},IntegerDigits[n]]] (* James C. McMahon, May 06 2025 *)

A136614(n) = A007953(a(n)) = A007953(A108773(n)).

A231963 Concatenate n with its UPC check digit, a(n) = 10*n + A237042(n).

Original entry on oeis.org

17, 24, 31, 48, 55, 62, 79, 86, 93, 109, 116, 123, 130, 147, 154, 161, 178, 185, 192, 208, 215, 222, 239, 246, 253, 260, 277, 284, 291, 307, 314, 321, 338, 345, 352, 369, 376, 383, 390, 406, 413, 420, 437, 444, 451, 468, 475, 482, 499, 505, 512, 529, 536, 543, 550, 567, 574, 581

Offset: 1

Alonso del Arte, Nov 15 2013

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.
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).
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.
Some manufacturers that have or might have UPCs ending in the numbers shown above include H. J. Heinz (013000) and the Hershey Company (068000).
As a rule of thumb, the terms mostly advance in steps of 7, sometimes 17.

			a(13) = 130 because 1 * 1 + 3 * 3 = 10, giving a check digit of 0.
a(14) = 147 because 1 * 1 + 4 * 3 = 13, and -13 = 7 mod 10.
a(15) = 154 because 1 * 1 + 5 * 3 = 16, and -16 = 4 mod 10.

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

GS1, Check Digit Calculator.
Eric Weisstein's World of Mathematics, UPC.

Cf. A237042, A093018, A108773.

a(n) = 10*n + vecsum(digits(n,100)*31\-10) % 10; \\ Kevin Ryde, Oct 03 2023

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.

A302801 a(n) is the number whose digits result from the concatenation of the digits of n and the digits of the product of the digits of n.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 200, 212, 224, 236, 248, 2510, 2612, 2714, 2816, 2918, 300, 313, 326, 339, 3412, 3515, 3618, 3721, 3824, 3927, 400, 414, 428, 4312, 4416, 4520, 4624, 4728, 4832, 4936, 500, 515

Offset: 0

Michel Marcus, Apr 13 2018

			For n=10, the digits of n are 1 and 0, their product is 0, and the concatenation gives 1 0 0.

Altug Alkan, Line graph of first differences of a(n) for n <= 10^5

Cf. A007954, A108773, A302575.

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[Product[Part[IntegerDigits[n],k],{k,Length[IntegerDigits[n]]}]]]],{n,0,51}] (* Stefano Spezia, Aug 13 2023 *)

mydigits(n) = if (n, digits(n), [0]);
a(n) = my(d = mydigits(n)); fromdigits(concat(d, mydigits(prod(k=1, #d, d[k]))));

cp's OEIS Frontend

A136614 Sum of digits of A108773 and of A136613.

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A108203 Numbers n put into lexicographical order which are the concatenation of k and the sum of the digits of k.

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A136613 Concatenation of (sum of digits of n) and n.

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A231963 Concatenate n with its UPC check digit, a(n) = 10*n + A237042(n).

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A302801 a(n) is the number whose digits result from the concatenation of the digits of n and the digits of the product of the digits of n.

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PARI