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A048869 Numbers for which reduced residue system contains as many primes as nonprimes. 2
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

There are no terms larger than 10890; it suffices to check to 52024. [Charles R Greathouse IV, Dec 19 2011]

LINKS

Table of n, a(n) for n=1..29.

EXAMPLE

n=45, phi(45)=24 and the reduced residue system mod 45 contains 12 primes {2,7,11,13,17,19,23,29,31,37,41,43} and 12 nonprimes {1,4,8,14,16,22,26,28,32,34,38,44}.

MATHEMATICA

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

PROG

(PARI) p=0; for(n=1, 6e4, if(isprime(n), p++); if(p==eulerphi(n)/2+omega(n), print1(n", "))) \\ Charles R Greathouse IV, Dec 19 2011

CROSSREFS

A000720(n)-A001221(n) = A000010(n) - [ A000720(n)-A001221(n) ].

Cf. A048597-A002110.

Sequence in context: A026438 A026442 A307712 * A039051 A047564 A154536

Adjacent sequences:  A048866 A048867 A048868 * A048870 A048871 A048872

KEYWORD

nonn,fini,full

AUTHOR

Labos Elemer

EXTENSIONS

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006

STATUS

approved

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)