Digit Sum

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The sum of the base-10 digits of an integer, used to generate or recognize sequences.

The digit sum s(n) adds up the base-10 digits of n. For example, s(1729) = 1 + 7 + 2 + 9 = 19. Digit-sum patterns commonly appear in two ways: (a) sequences where each term is the digit sum of the previous term, and (b) sequences whose nth term relates to s(n) somehow. The digit sum is invariant under permutation of digits and under multiples of 9 (a number is divisible by 9 iff its digit sum is).

s(n) = sum of base-10 digits of n

• 1729 → 1+7+2+9 = 19
• s(1) = 1, s(10) = 1, s(100) = 1 (all 1s)
• 144 → 1+4+4 = 9 (divisible by 9 ↔ digit sum divisible by 9)

How to recognize it

• Suspect digit-sum rules when a sequence resets or loops between small values (1–9)
• Check divisibility by 3 or 9 via the digit sum
• If consecutive terms relate via summed digits, iterate the rule to confirm

Common mistakes

• Treating digit sum as digit product (9 → 9 but 19 → 9, not 10)
• Applying digit sum in a non-decimal base without adjusting
• Forgetting that multi-digit input numbers can be reduced repeatedly to a single digit (digital root)

Practice guidance

Number Sequences