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Finite sequences

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Finite sequences are sequences with a provably finite number of terms, although not only the last term might be unknown but even the number of terms might be unknown, and which therefore have a last term (which is the largest term if the sequence is [‍strictly or nonstrictly‍] increasing, and which is the smallest term if the sequence is [‍strictly or nonstrictly‍] decreasing). For example, T. D. Noe has demonstrated that there are precisely 449 numbers that are both superabundant and highly composite (A166981).

In the OEIS, finite sequences are assigned the keyword fini.

Extremely long finite sequences


Provably finite sequences of unknown length


Conjectured finite sequences

A conjectured finite sequence is ultimately either

  • a finite sequence (the conjecture is proved true);
  • a sequence for which the finiteness of the cardinality (finite or countably infinite) is proved undecidable;
  • an infinite sequence (the conjecture is proved false).

A finite sequence is finite regardless of whether or not we know it to be finite, but a sequence can only be declared as finite if it has been proved, in which case the sequence is assigned keyword:fini in the OEIS.

It has been conjectured the the number of Fermat primes (A019434) is finite (e.g. 5 terms). There is a similar conjecture for generalized Fermat primes of any given specific form.

See also