Smith Number Checker

How it is calculated

A Smith number is a composite whole number whose digit sum equals the sum of the digit sums of its prime factors, each factor counted as often as it divides the number. The tool factorises n by trial division, adds the digits of n to get S(n), adds the digits of every prime factor (with repeats) to get S(factors), and answers Yes only if S(n) = S(factors) and n is not itself prime. A prime always answers No, because its only factor is the number itself; for 13 both sums are 4, yet the answer is still No. The input must be a whole number from 2 to 1,000,000,000; decimals, negatives, 0, 1, numbers above one billion and text such as 1,000 give no result. Digit sums are base 10 only.

Common mistakes

• Expecting a prime such as 13 to qualify: its digit sum matches its single factor's, but primes are excluded by definition, so the answer is No.
• Adding each distinct prime factor only once: 27 = 3 × 3 × 3 needs three 3s (sum 9), and leaving out repeats gives the wrong factor sum.
• Adding the factors themselves instead of their digits: 2 + 3 + 3 + 37 = 45 is not the figure to compare, 18 is.
• Entering a decimal, a negative, 1, or a number with separators such as 1,000: the tool accepts only plain whole numbers from 2 up to one billion and shows no result otherwise.
• Checking the number written in another base rather than plain base-10 digits.