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A046642
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Numbers k such that k and number of divisors d(k) are relatively prime.
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18
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1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131
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OFFSET
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1,2
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COMMENTS
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Numbers k such that tau(k)^phi(k) == 1 (mod k), where tau(k) is the number of divisors of k (A000005) and phi(k) is the Euler phi function (A000010). - Michel Lagneau, Nov 20 2012
Density is at least 4/Pi^2 = 0.405... since A056911 is a subsequence, and at most 1/2 since all even numbers in this sequence are squares. The true value seems to be around 0.4504. - Charles R Greathouse IV, Mar 27 2013
They are called anti-tau numbers by Zelinsky (see link) and their density is at least 3/Pi^2 (theorem 57 page 15). - Michel Marcus, May 31 2015
Spiro (1981) proved that the number of terms of this sequence that do not exceed x is c * x + O(sqrt(x)*log(x)^3), where 0 < c < 1 is the asymptotic density of this sequence.
The odd numbers whose number of divisors is a power of 2 (the odd terms of A036537) are terms of this sequence. Their asymptotic density is A327839/A076214 = 0.4212451116... which is a better lower bound than 4/Pi^2 for the asymptotic density of this sequence.
A better upper limit than 0.5 can be obtained by considering the subsequence of odd numbers whose 3-adic valuation is not of the form 3*k-1 (i.e., odd numbers without those k with gcd(k, tau(k)) = 3), whose asymptotic density is 6/13 = 0.46153...
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 49, 459, 4535, 45145, 450710, 4504999, 45043234, 450411577, 4504050401, ... (End)
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LINKS
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FORMULA
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MATHEMATICA
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PROG
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(Haskell)
a046642 n = a046642_list !! (n-1)
a046642_list = map (+ 1) $ elemIndices 1 a009191_list
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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