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Search: id:A074987
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| A074987 |
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a(n) = the least m not equal to n such that phi(m) = phi(n). |
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+0
1
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| 2, 1, 4, 3, 8, 3, 9, 5, 7, 5, 22, 5, 21, 7, 16, 15, 32, 7, 27, 15, 13, 11, 46, 15, 33, 13, 19, 13, 58, 15, 62, 17, 25, 17, 39, 13, 57, 19, 35, 17, 55, 13, 49, 25, 35, 23, 94, 17, 43, 25, 64, 35, 106, 19, 41, 35, 37, 29, 118, 17, 77, 31, 37, 51, 104, 25, 134, 51, 92, 35, 142
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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In 1922, Carmichael asked if for any given natural number n there exists a natural number m different from n such that phi(m) = phi(n). A. Shalafly and S. Wagon showed in 1994 that if there is an n such that phi(m) != phi(n) for all m distinct from n, then n must be greater than 10^(10^7). (Tattersall)
I conjecture that a(n) <= 2n. I have checked this for all n <= 10^4. (It is not possible to do better than the 2n upper bound since a(11) = 2*11.)
For odd n the conjecture is true because phi(n)=phi(2n). - T. D. Noe, Oct 18 2006
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REFERENCES
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Tattersall, J. "Elementary Number Theory in Nine Chapters", Cambridge University Press, 2001, pp. 162-163.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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EXAMPLE
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phi(5) = 4 and 8 is the least natural number k different from 5 such phi(k) = 4. Hence phi(5) = 8.
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MATHEMATICA
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l = {}; Do[ e = EulerPhi[n]; i = 1; While[e != EulerPhi[i] || n == i, i++ ]; l = Append[l, i], {n, 1, 100}]; l
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CROSSREFS
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Sequence in context: A058354 A085930 A087207 this_sequence A128280 A106625 A008347
Adjacent sequences: A074984 A074985 A074986 this_sequence A074988 A074989 A074990
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Oct 02 2002
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