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A060771
Upper ends of record prime gaps under consideration of the prime number theorem.
1
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.
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
Sequence in context: A167895 A106712 A019391 * A265788 A061245 A361898
KEYWORD
nonn
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
STATUS
approved