Calculators
Sylvester Number Calculator
Calculate exact Sylvester sequence terms for indices 0 to 12 privately in your browser, with every digit preserved.
Calculate a term of the Sylvester sequence, starting at S(0) = 2. Enter an index from 0 to 12. Exact calculations stay in your browser.
Explore the rapidly growing Sylvester sequence.
The Sylvester sequence starts with S(0) = 2. Each later term is one plus the product of all preceding terms: S(n) = 1 + S(0) × … × S(n − 1). Equivalently, S(n + 1) = S(n) × (S(n) − 1) + 1. The first terms are 2, 3, 7, 43, 1807 and 3263443. For example S(3) = 2 × 3 × 7 + 1 = 43. Every pair of distinct terms is coprime, but terms need not be prime. The reciprocals form an Egyptian fraction expansion of 1: the sum from index 0 through n is 1 − 1 / (S(n + 1) − 1). This calculator uses zero-based indexing and accepts 0 through 12, using up to two digits. Leading zeros and surrounding whitespace are accepted; signs, decimals and scientific notation are rejected. The upper limit keeps the rapidly growing output manageable. Exact BigInt arithmetic preserves every result digit, and results include the decimal digit count. Inputs are not uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What are the first Sylvester numbers?
Starting at index zero, the sequence begins 2, 3, 7, 43, 1807 and 3263443.
How is each term calculated?
Each term after the first is one plus the product of all preceding terms. Equivalently, multiply the previous term by one less than itself, then add one.
Are all Sylvester numbers prime?
No. Distinct terms are pairwise coprime, but individual terms can be composite. For example 1807 = 13 × 139.
Why is the index limited to 12?
The sequence grows roughly doubly exponentially. The limit keeps exact browser results manageable. Calculations use BigInt and stay on your device.