Let be the simple continued fraction of a "generic" real number, where the numbers are the partial quotients. Then the Khinchin (or Khintchine) harmonic mean
(1)
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defined analogously to the Khinchin constant but with the partial quotients taken to the power, exists and has a unique common value (except for a set of real numbers with measure zero) given by
(2)
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(3)
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(4)
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(OEIS A087491; Bailey et al. 1997, Plouffe).
Khinchin's constant and the Khinchin harmonic mean are just two of an infinite family of such constants , the first few of which are summarized in the following table.
OEIS | value | |
0 | A002210 | 2.685452001065306445309714835481795693820382293994462 |
A087491 | 1.745405662407346863494596309683661067294936618777984 | |
A087492 | 1.450340328495630406052983076680697881408299979605904 | |
A087493 | 1.313507078687985766717339447072786828158129861484792 | |
A087494 | 1.236961809423730052626227244453422567420241131548937 | |
A087495 | 1.189003926465513154062363732771403397386092512639671 | |
A087496 | 1.156552374421514423152605998743410046840213070718761 | |
A087497 | 1.133323363950865794910289694908868363599098282411797 | |
A087498 | 1.115964408978716690619156419345349695769491182230400 | |
A087499 | 1.102543136670728013836093402522568351022221284149318 | |
A087500 | 1.091877041209612678276110979477638256493272651429656 |