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Search: id:A048869
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| A048869 |
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Numbers for which reduced residue system contains as many primes as nonprimes. |
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+0
3
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| 3, 4, 5, 6, 7, 9, 10, 15, 21, 45, 58, 82, 86, 92, 105, 116, 196, 238, 308, 310, 320, 380, 972, 978, 996, 1068, 1368, 5640, 10890
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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This sequence is finite, since the number of primes < n is ~ n/log(n), but liminf phi(n) / ( n*log(log(n)) ) = exp(-gamma), a consequence of Mertens's theorem (see Hardy and Wright's Theory of Numbers). Also, if there exists a further element, it is >700000 (as verified with the enclosed Mathematica code). (Question: is it possible to show that there are no further such elements by using explicit bounds in the Prime Number Theorem and in Mertens's theorem?) - Reiner Martin (reinermartin(AT)hotmail.com), Jan 16 2002
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EXAMPLE
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n=45,Phi(45)=24 reduced residue system (45) has a prime-clear part :{2,7,11,13,17,19,23,29,31,37,41,43} and a nonprime set (including 1) of equal size:{1,4,8,14,16,22,26,28,32,34,38,44}.
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MATHEMATICA
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Select[Range[700000], 2(PrimePi[ # ] - Length[FactorInteger[ # ]]) == EulerPhi[ # ]&]
For[i = 1, i < 100000000000, i++, If[2(PrimePi[i] - Length[FactorInteger[i]]) == EulerPhi[i], Print[i]]]; - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006
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CROSSREFS
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A000720(n)-A001221(n) = A000010(n) - [ A000720(n)-A001221(n) ]. Cf. A048597, ..., A002110.
Sequence in context: A083121 A026438 A026442 this_sequence A039051 A047564 A154536
Adjacent sequences: A048866 A048867 A048868 this_sequence A048870 A048871 A048872
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KEYWORD
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more,nonn
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu)
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EXTENSIONS
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More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006
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