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Chapter 2 in the Aigner and Ziegler book is devoted to Bertrand's postulate. The proof given starts by showing Bertrand's postulate is true just for n < 4000.

After 2, each prime is less than twice the previous prime. So, even if these were the only primes up to 4002, Bertrand's postulate would still be true for the specified range.

However, these are different from the Bertrand primes (A006992) after 2503, as that sequence requires the very largest prime smaller than twice the previous one, since twice 2503 is 5006 and 5003 is the largest prime less than that.

Erdős Pál used this sequence, with 4001 instead of 5003, in his 1932 proof of Bertrand's postulate, attributing it to Edmund Landau ("einer Bemerkung des Herrn Landau"), which Aigner and Ziegler refer to as "Landau's trick" in their book.

RECORDS transform of A080189; prime p sets a new record for the number of applications of f that are required to reach 2. - a(n) = prime preceding 2*a(n-1) as long as a(n-1) is a term of A080191; if however a(n-1) is a term of A080192, then a(n) > 2*a(n-1). - Next term a(32) > 3932600000, presumably a(32) = 5274863189, a(33) = 10549726367. - The sequence coincides with A006992 (Bertrand primes: a(n) is largest prime < 2*a(n-1)) for the first 17 terms; first divergence occurs after term 39869 because this is the first term which belongs to A080192.

The sequence is connected with our sieve selecting the primes of the union of {2,3} and A164333 from all primes.