# Sum of squared digits returns incorrect result

For the sake of practicing writing code, I've given myself an exercise in writing a script that checks whether a given positive integer is happy, i.e. take the digits of a number, square them, and add them together. If the sum is 1, then the integer is happy; otherwise, continue this process with the new sum.

While checking to make sure the first step is carried out correctly, I run into a problem. Clearly, n = 1 is happy, yet determining this with a defined function gives the wrong conclusion:

 In[1]:= ssd[x_] = Total[IntegerDigits[x]^2];
(* ssd = Sum of Squared Digits *)
ssd[1]
Total[IntegerDigits[1]^2]

Out[2]:= {3}

Out[3]:= 1


This also happens for other integers larger than 1.

 In[4]:= Table[ssd[x], {x, 1, 9}] // Flatten
Table[Total[IntegerDigits[x]^2], {x, 1, 9}]

Out[4]:= {3, 4, 5, 6, 7, 8, 9, 10, 11}

Out[5]:= {1, 4, 9, 16, 25, 36, 49, 64, 81}


Is there a bug here, or am I doing something wrong? (I'm using 10.4.1 for Windows, if that makes any difference.)

• use := (SetDelayed) instead of = (Set) when you define ssd; i.e., try ssd[x_] := Total[IntegerDigits[x]^2]
– kglr
Commented Jan 12, 2017 at 19:57
• to see why you are getting {3} inspect the output of Trace[Total[IntegerDigits[x]^2] /. x -> 1 ]
– kglr
Commented Jan 12, 2017 at 20:00
• Huh, any idea where that 2 is coming from? Commented Jan 12, 2017 at 20:06
• using FullForm[IntegerDigits[x]^2]  we see that FullForm of IntegerDigits[x]^2 is Power[IntegerDigits[x], 2]. Then, since Total works with any head:, ie,.Total[foo[a, b, c]] is a+b+c, Total[Power[IntegerDigits[x], 2]] gives 2+IntegerDigits[x]
– kglr
Commented Jan 12, 2017 at 20:22

The fix: use SetDelayed instead of Set when defining ssd:

 ssd2[x_] := Total[IntegerDigits[x]^2]
ssd2[1]


1

Why does ssd[1] give{3}?

For

ssd[x_] = Total[IntegerDigits[x]^2];


To see how {3} is obtained use Trace

ssd[1] // Trace


{ssd[1], 2 + IntegerDigits[1], {IntegerDigits[1], {1}}, 2+{1}, {3}}

As you see ssd[1] is stored as 2 + IntegerDigits[1], More generally, with a symbolic input:

ssd[z]


2 + IntegerDigits[z]

This is because (see SetDelayed >> Properties and Relations )

The right side of an immediate definition is evaluated when the definition is made

How does Total[IntegerDigits[x]^2] become 2 + IntegerDigits[x]?

Inspecting the FullForms of subexpressions on the rhs of your definition:

IntegerDigits[x]^2 // FullForm


Power[IntegerDigits[x], 2]

and

Total[IntegerDigits[x]^2] // FullForm


Plus[2, IntegerDigits[x]]

The reason for this is because (see Total >> Properties and Relations):

Total[list] is equivalent to Apply[Plus,list]

That is, Total[expr] replaces the head of expr with Plus. Hence

Total[foo[a, b, c]]


a + b + c

and

Total[Power[a, b]]


a + b

Thus,

Total[Power[IntegerDigits[x], 2]]


2 + IntegerDigits[x]