sigma4214

Sigma4214 is a term used in number theory to denote the sum-of-divisors function evaluated at the integer 4214, i.e., sigma(4214). The sum-of-divisors function, sigma(n), adds all positive divisors of n and is multiplicative: if gcd(a,b) = 1, then sigma(ab) = sigma(a) sigma(b). For prime powers, sigma(p^k) = (p^{k+1} − 1)/(p − 1).

The integer 4214 factors as 2 × 7^2 × 43. Using multiplicativity, sigma(4214) = sigma(2) sigma(7^2) sigma(43) = (1 + 2) × (1 + 7 + 49) × (1 + 43) = 3 × 57 × 44 = 7524. Thus the sum of all positive divisors of 4214 is 7524.

The sum of proper divisors (excluding 4214 itself) is 7524 − 4214 = 3310, which is less than 4214. Therefore 4214 is a deficient number. The total number of positive divisors is (1 + 1)(2 + 1)(1 + 1) = 12.

Sigma4214 thus serves as a concrete example of applying prime factorization and the divisor-sum function to compute sigma(n). See also the divisor function and arithmetic functions related to divisor sums.