Sylvester’s sequence
Content created by Egbert Rijke, Fredrik Bakke and Louis Wasserman.
Created on 2024-10-25.
Last modified on 2025-08-30.
module elementary-number-theory.sylvesters-sequence where
Imports
open import elementary-number-theory.natural-numbers open import elementary-number-theory.ordinal-induction-natural-numbers open import elementary-number-theory.products-of-natural-numbers open import univalent-combinatorics.standard-finite-types
Idea
Sylvester’s sequence¶
is the sequence s
of
natural numbers in which s n
is
the successor of the
product of all the
numbers s i
for i < n
, i.e.,
The first few entries in this sequence are s 0 = 2
, s 1 = 3
, s 2 = 7
, and
s 3 = 43
.
Sylvester’s sequence is listed as A000058 in the OEIS [OEIS].
Definitions
sylvesters-sequence-ℕ : ℕ → ℕ sylvesters-sequence-ℕ = ordinal-ind-ℕ ( λ _ → ℕ) ( λ n f → succ-ℕ (Π-ℕ n (λ i → f (nat-Fin n i) (strict-upper-bound-nat-Fin n i))))
References
- [OEIS]
- OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences. website. URL: https://oeis.org.
External links
- Sylvester’s sequence at Wikidata
Recent changes
- 2025-08-30. Louis Wasserman and Fredrik Bakke. Move sequences to
lists
namespace (#1476). - 2024-10-25. Egbert Rijke. Sylvester’s sequence (#1210).