A060771 Upper ends of record prime gaps under consideration of the prime number theorem.

3, 5, 7, 11, 29, 97, 127, 541, 907, 1151, 1361, 15727, 19661, 31469, 156007, 360749, 370373, 1357333, 2010881, 17051887, 20831533, 47326913, 191913031, 436273291, 2300942869, 3842611109, 4302407713, 10726905041, 22367085353, 25056082543

Offset: 1

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn

Comments

Every element > 7 must be in A000101 too (consider the derivatives of x/log(x) to prove this), but not conversely. The sequence is infinite since lim sup (length of n-th prime gap/log(n-th prime)) is infinite, proved by Westzynthius, see Ribenboim.

Examples

			541 is okay since 541/log(541) - 523/log(523) = 2.4108.. was not reached by smaller primes
		

References

• P. Ribenboim, The Book of Prime Number Records, Chapter about prime gaps.
• E. Westzynthius, Über die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind Comm. Phys. Math. Helsingfors 25, 1931.

Crossrefs

Cf. A060769, A000101.

Formula

A prime p belongs to the sequence iff p/log(p) - q/log(q) attains a new high, where q is the preceding prime.