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A065557
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Solutions k of the equation phi(k) = phi(k-1) + phi(k-2). Also known as Phibonacci numbers.
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17
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3, 5, 7, 11, 17, 23, 37, 41, 47, 101, 137, 233, 257, 857, 1037, 1297, 1541, 1601, 2017, 4337, 6527, 9179, 14401, 16097, 30497, 55387, 61133, 62801, 65537, 72581, 77617, 110177, 152651, 179297, 244967, 299651, 603461, 619697, 686737, 1876727
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OFFSET
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1,1
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COMMENTS
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All terms listed here are squarefree. (Thanks to Vladeta Jovovic for this observation.) The first two nonsquarefree terms are 72340252337 and 179115011177. There are 205 terms < 5*10^11. Most early terms are prime but later most terms are composite. - Jud McCranie, Feb 21 2012
Bagers (1981) named these numbers Phibonacci numbers and asked about the existence of composite terms. According to the solution, P. J. Weinberg found 70 terms below 2*10^8, of which 46 are composite. The existence of an even term was discussed, and if it exists, it exceeds 10^1600. - Amiram Eldar, Mar 01 2020
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover Pub., NY, 1966.
Stanley J. Bezuszka and Margaret Kenney, Number Treasury: A Sourcebook of Problems for Calculators and Computers, Dale Seymour Publications, 1982, pp. 126 and 179.
Mihai Caragiu, Sequential Experiments with Primes, Springer, 2017, chapter 4, p. 152.
Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 117, entry 1037.
József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 224.
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LINKS
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Anders Bagers, Problem E2833, The American Mathematical Monthly, Vol. 87, No. 5 (1980), p. 404, solution, ibid., Vol. 88, No. 8 (1981), p. 622.
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EXAMPLE
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phi(23) = phi(22) + phi(21) (22=10+12), so 23 is in the sequence.
phi(101) = phi(100) + phi(99) (100=40+60), so 101 is in the sequence.
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MATHEMATICA
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Select[ Range[3, 10^6], EulerPhi[ # ] == EulerPhi[ # - 1] + EulerPhi[ # - 2] & ]
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PROG
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(PARI): for(n=3, 10^8, if(eulerphi(n)==eulerphi(n-1)+eulerphi(n-2), print1(n, ", ")))
(PARI) { n=0; e1=eulerphi(2); e2=eulerphi(1); for (m=3, 10^9, e=eulerphi(m); if (e==e2 + e1, write("b065557.txt", n++, " ", m); if (n==100, return)); e2=e1; e1=e ) } \\ Harry J. Smith, Oct 22 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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