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A032358
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Number of iterations of phi(n) needed to reach 2.
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4
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0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 3, 4, 4, 5, 3, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 5, 3, 5, 4, 4, 4, 5, 4, 5, 4, 4, 5, 5, 4, 5, 5, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,4
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COMMENTS
| This sequence is additive (but not completely additive). [Charles R Greathouse IV, Oct 28 2011]
Shapiro asks for a proof that for every n > 1 there is a prime p such that a(p) = n. [Charles R Greathouse IV, Oct 28 2011]
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REFERENCES
| Harold Shapiro, An arithmetic function arising from the phi function, Amer. Math. Monthly 50:1 (1943), pp. 18-30.
Catlin: Concerning the iterated phi-function. Amer Math. Monthly 77 (1970), pp. 60-61.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
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FORMULA
| a(n) = a(phi(n))+1, a(1) = -1. - Vladeta Jovovic, Apr 29 2003
a(n) = A003434(n) - 1 = A049108(n) - 2.
Contribution from Charles R Greathouse IV, Oct 28 2011
(Start)
Shapiro proves that log_3(n/2) <= a(n) < log_2(n) and also
a(mn) = a(m) + a(n) if either m or n is odd and a(mn) = a(m) + a(n) + 1 if m and n are even.
(End)
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MATHEMATICA
| Table[Length[NestWhileList[EulerPhi[#]&, n, #>2&]]-1, {n, 3, 80}] (* From Harvey P. Dale, May 01 2011 *)
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PROG
| (Haskell)
a032358 = length . takeWhile (/= 2) . (iterate a000010)
-- Reinhard Zumkeller, Oct 27 2011
(PARI) a(n)=my(t); while(n>2, n=eulerphi(n); t++); t \\ Charles R Greathouse IV, Oct 28 2011
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CROSSREFS
| Cf. A000010.
Sequence in context: A083552 A176835 A078704 * A011960 A187035 A008615
Adjacent sequences: A032355 A032356 A032357 * A032359 A032360 A032361
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KEYWORD
| nice,nonn,easy
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AUTHOR
| Ursula Gagelmann (gagelmann(AT)altavista.net)
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EXTENSIONS
| a(2) = 0 added and offset adjusted, suggested by David W. Wilson.
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