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Search: id:A060771
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| A060771 |
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Upper ends of record prime gaps under consideration of the prime number theorem. |
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+0
2
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| 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
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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P. Ribenboim, The Book of Prime Number Records, Chapter about prime gaps.
E. Westzynthius, Ueber die Verteilung der Zahlen, die zu den n ersten Primzahlen teilerfremd sind Comm. Phys. Math. Helsingfors 25, 1931.
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FORMULA
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A prime p belongs to the sequence iff p/log(p) - q/log(q) attains a new high, where q is the preceding prime.
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EXAMPLE
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541 is okay since 541/log(541) - 523/log(523) = 2.4108.. was not reached by smaller primes
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CROSSREFS
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Cf. A060769, A000101.
Sequence in context: A142247 A106712 A019391 this_sequence A061245 A137355 A147143
Adjacent sequences: A060768 A060769 A060770 this_sequence A060772 A060773 A060774
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KEYWORD
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nonn
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AUTHOR
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Ulrich Schimke (ulrschimke(AT)aol.com)
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