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Digit Sum

A digit sum is a sum of the base- digits of , which can be implemented in the Wolfram Language as DigitSum[n_, b_:10] := Total[IntegerDigits[n, b]]The following table gives for , 2, ... and small .OEIS for , 2, ...2A0001201, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, ...3A0537351, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, ...4A0537371, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, ...5A0538241, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, ...6A0538271, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, ...7A0538281, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, ...8A0538291, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, ...9A0538301, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, ...10A0079531, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, ...Plotting versus and gives the plot shown above.The digits sum satisfies the congruence(1)In base 10, this congruence is the basis of casting out nines and of fast divisibility tests such as those for 3 and 9. satisfies the following unexpected identity(2)the case of which was given in the 1981 Putnam competition..

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