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

1,1

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.

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.

Links

Table of n, a(n) for n=1..30.

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.

Example

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

Crossrefs

Cf. A060769, A000101.

Sequence in context: A167895 A106712 A019391 * A265788 A061245 A361898

Adjacent sequences: A060768 A060769 A060770 * A060772 A060773 A060774

Keyword

nonn

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Status

approved