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function construction - Error computing sum of sum of digits



I've defined a function that computes the sum of the base-b digits of n:


DigitSum[n_, b_] := Total[IntegerDigits[n, b]]

Then I defined a function that computes the sum of the base-b digits of all of the integers up to x:


CumDigitSum[x_, b_] := Sum[DigitSum[n, b], {n, 1, x}]

Using these functions, I get


CumDigitSum[1000000,10]=27000001

which is correct. But then for larger inputs I get nonsense like



CumDigitSum[1000001,10]=500011500011

If I work in a different base, the same thing happens: at exactly 1000001, Mathematica begins computing the sum incorrectly. If I bypass my user-defined functions and just write what I mean, I get the correct answer:


Sum[Total[IntegerDigits[n, 10]], {n, 1, 1000001}] = 27000003

Any idea what could be happening here?



Answer



If you wish to compute the correct values using the method you have chosen you could specify Method -> "Procedural" for Sum:


CumDigitSum[x_, b_] := Sum[DigitSum[n, b], {n, 1, x}, Method -> "Procedural"]


CumDigitSum[1000001, 10]


27000003

However, the problem comes form the fact that Sum attempts to speed the calculation by finding a symbolic equivalent. Let's see what your DigitSum returns with symbolic input:


DigitSum[n, b]


b + n


That's clearly not correct generally! Why does this happen? Mathematica functions often work on arbitrary expressions as well as lists, and that is the case with Total. First the IntegerDigits call remain unevaluated:


IntegerDigits[n, b]


IntegerDigits[n, b]

But then Total adds its arguments as though this were {n, b}:


IntegerDigits[n, b] // Total



b + n

To prevent this you can add a Condition or PatternTest as belisarius recommended in a comment above:


DigitSum[n_?NumericQ, b_] := Total[IntegerDigits[n, b]]

This will block Sum form finding a (false) symbolic equivalent thereby forcing it to use a procedural evaluation, even with your original CumDigitSum definition.


For this particular case it is somewhat cleaner to use Tr in place of Total as it will not sum the arguments of an arbitrary expression:


Tr[IntegerDigits[n, b]]



Tr[IntegerDigits[n, b]]

Therefore:


DigitSum[n_, b_] := Tr[IntegerDigits[n, b]]

By the way, it is not recommended to start user Symbol names with capital letters as these can conflict with built-in functions, now or later.


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