Function Repository Resource:

AdditiveDigitalRoot

Source Notebook

Compute the additive digital root, in any base, of an integer

Contributed by: Christopher Stover

ResourceFunction["AdditiveDigitalRoot"][n,b]

gives the additive digital root of the non-negative integer n when expressed in the base b.

ResourceFunction["AdditiveDigitalRoot"][n]

gives the base-10 additive digital root of n.

Details

Starting with a non-negative integer n, add its base-b digits, then add the digits of the resulting number, etc., until the result has only one digit. The one-digit integer marking the stopping point of this process is called the additive digital root of n.

As an example in base-10, consider n=1191:

Adding its digits yields 1+1+9+1=12.

Adding the digits of the result yields 1+2=3. Because 3 is a single-digit number, the process stops.

Because the integer marking the termination of the process is 3, ResourceFunction["AdditiveDigitalRoot"][1191] returns 3.

The additive digital root is sometimes called the digital root (sans "additive"), and is denoted dr_b(n). It is also sometimes referred to as the repeated digital sum of n.

ResourceFunction["AdditiveDigitalRoot"] threads elementwise over lists.

Examples

Basic Examples (2)

Compute the additive digital root of 1191:

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Compute the additive digital root of 182, base-14 (note that the result has two digits base-10, but only a single base-14 digit):

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Scope (2)

AdditiveDigitalRoot threads elementwise over lists:

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Compute the base-10 additive digital roots of the first 100 integers (OEIS A010888):

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Properties and Relations (5)

The additive digital root can be computed iteratively using NestWhile:

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$With[{n = RandomInteger[10^9], b = RandomInteger[{2, 20}]}, ResourceFunction["AdditiveDigitalRoot"][n, b] == NestWhileList[Plus @@ IntegerDigits[#, b] &, n, # >= b &][[-1]]]$

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The additive digital root can be computed using Floor:

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$With[{n = RandomInteger[10^9], b = RandomInteger[{2, 20}]}, ResourceFunction["AdditiveDigitalRoot"][n, b] == n - (b - 1) Floor[(n - 1)/(b - 1)]]$

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There is a well-known closed formula for the additive digital root in terms of Mod:

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$With[{n = RandomInteger[10^9], b = RandomInteger[{2, 20}]}, ResourceFunction["AdditiveDigitalRoot"][n, b] == Mod[n, b - 1, 1]]$

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The number of iterations required to reach the end of the digit addition process is called the additive persistence and is returned by the resource function AdditivePersistence:

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The multiplicative analogue of the additive digital root is called the multiplicative digital root and is returned by the resource function MultiplicativeDigitalRoot:

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