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A341749
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Numbers k such that gcd(k, phi(k)) > log(log(k)).
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2
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96
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OFFSET
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1,2
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COMMENTS
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First differs from A080197 at n = 28.
Erdős et al. (2008) proved that the asymptotic density of numbers k such that gcd(k, phi(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).
There are only 8 cyclic numbers (A003277) in this sequence: 1, 2, 3, 5, 7, 11, 13, 15. All the other terms are in A060679. The first term of A060679 which is not in this sequence is 1622.
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LINKS
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EXAMPLE
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16 is a term since gcd(16, phi(16)) = gcd(16, 8) = 8 > log(log(16)) = 1.0197...
17 is not a term since gcd(17, phi(17)) = gcd(17, 16) = 1 < log(log(17)) = 1.0414...
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MATHEMATICA
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Select[Range[100], GCD[#, EulerPhi[#]] > Log[Log[#]] &]
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PROG
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(PARI) isok(k) = (k==1) || (gcd(k, eulerphi(k)) > log(log(k))); \\ Michel Marcus, Feb 19 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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