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# A000000

The empty sequence.

## Contents

1,1

### Comment

• The A-number A000000 is inadmissible in the main OEIS (it might have been used for the empty sequence, except that since that sequence has no terms, the lookup programs would not be able to handle it).
• It is very fitting for the empty sequence to be sequence A000000 since it is the only sequence with cardinality 0.
• The empty sequence as there is only one empty sequence, even though it is the “solution” (so to speak) of a countably infinite number of unsatisfiable sequence definitions (or enunciable problems without solutions).
• Any sequence whose definition is not satisfiable results in the empty sequence.
• We may consider sequences whose definition are conjectured not satisfiable (sequences conjectured empty) and sequences whose definition are proved not satisfiable (sequences proved empty).

### Example

• Even numbers  n   ≥   6
which are not the sum of at most 2 odd primes. (Cf. “strong” Goldbach conjecture.)
• Odd numbers  n   ≥   9
which are not the sum of at most 3 odd primes. (Cf. “weak” Goldbach conjecture.)
• Positive integers  n
such that  n  k = a  k + b  k, k   ≥   3, a > 0, b > 0
. (Cf. Andrew Wiles’ proof (final corrected proof published in 1995) of Fermat’s last theorem (proposed in 1637, proof never found).)
• Positive integers  n
that are not the sum of at most  k
(not neccessarily distinct)  k
-gonal numbers. (Cf. Cauchy’s proof (1813) of Fermat’s polygonal number theorem (proposed in 1638, proof never found).)

nonn,easy

### Author

_Daniel Forgues_, Jun 15, 2010

### Status

The A-number A000000 is inadmissible in the main OEIS (it might have been used for the empty sequence, except that since that sequence has no terms, the lookup programs would not be able to handle it).