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A051487
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Numbers n such that phi(n) = phi(n - phi(n)).
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5
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2, 6, 12, 24, 48, 96, 150, 192, 300, 384, 600, 726, 750, 768, 1200, 1452, 1500, 1536, 2310, 2400, 2904, 3000, 3072, 3174, 3750, 4620, 4800, 5046, 5808, 5874, 6000, 6090, 6144, 6348, 6930, 7500, 7986, 9240, 9600, 10086, 10092, 10374, 11550, 11616
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| This sequence is infinite, in fact 3*2^n is a subsequence because if m=3*2^n then phi(m-phi(m))=phi(3*2^n-2^n)=2^n=phi(m). Also, if p is a Sophie Germain prime greater than 3 then for each natural number n, 2^n*3*p^2 is in the sequence. Note that there exist terms of this sequence like 750 or 2310 that they aren't of either of these forms. - Farideh Firoozbakht, Jun 19 2005
If n is an even term greater than 2 in this sequence then 2n is also in the sequence. Because for even numbers m we have phi(2m)=2*phi(m) so phi(2n)=2*phi(n)=2*phi(n-phi(n)) and since n is an even number greater than 2 n-phi(n) is even so 2*phi(n-phi(n))=phi(2n-2*phi(n))=phi(2n-phi(2n)) hence phi(2n)=phi(2n-phi(2n)) and 2n is in the sequence. Conjecture: All terms of this sequence are even. - Farideh Firoozbakht, Jul 04 2005
If n is in the sequence and the natural number m divides gcd(n,phi(n)) then m*n is in the sequence. The facts that I have found about this sequence earlier (Jun 19 2005 and Jul 04 2005) are consequences of this. If p is a Sophie Germain prime greater than 3, k>1 and k & n are natural numbers then 2^n*3*p^k are in the sequence. - Farideh Firoozbakht, Dec 10 2005
Numbers n such that phi(n) = phi(n + phi(n)) includes all n = 2^k. - Jonathan Vos Post, Oct 25 2007
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REFERENCES
| R. K. Guy, Unsolved Problems in Number Theory B42.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
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MATHEMATICA
| Select[Range[11700], EulerPhi[ # ] == EulerPhi[ # - EulerPhi[ # ]] &] (Firoozbakht)
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PROG
| (PARI) isA051487(n) = eulerphi(n) == eulerphi(n - eulerphi(n)) [From Michael Porter, Dec 07 2009]
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CROSSREFS
| Cf. A005384, A051488.
Sequence in context: A054061 A118224 A003680 * A111286 A058295 A132176
Adjacent sequences: A051484 A051485 A051486 * A051488 A051489 A051490
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KEYWORD
| nonn,nice
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AUTHOR
| R. K. Guy (rkg(AT)cpsc.ucalgary.ca)
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu)
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