Function Repository Resource:

AdditiveDigitalRoot (1.0.0) current version: 1.1.0 »

Compute the additive digital root of an integer

Contributed by: Christopher Stover

ResourceFunction["AdditiveDigitalRoot"][n]

returns the additive digital root of n.

Details

Starting with a non-negative integer n, add its 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, 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 of a non-negative integer n is sometimes called its digital root (sans "additive"), which is denoted dr(n). It is also sometimes referred to as the repeated digital sum of n.

ResourceFunction["AdditiveDigitalRoot"] threads elementwise over lists.

Examples

Basic Examples (1)

Compute the additive digital root of 1191:

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Scope (2)

AdditiveDigitalRoot threads elementwise over lists:

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

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Properties and Relations (5)

The result returned by AdditiveDigitalRoot can be manually computed using NestWhileList:

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There is a well-known closed formula for the additive digital root:

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$With[{n = RandomInteger[{10^6, 10^9}]}, ResourceFunction["AdditiveDigitalRoot"][n] == 1 + Mod[n - 1, 9]]$

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The additive digital root of an integer can also be manually computed with no high-level functions:

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Because testint3 is a single-digit integer, the process terminates here, and because testint3 equals 1, the additive digital root of testint1=1234 is equal to 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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