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Search: id:A066426
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| A066426 |
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Conjectured values for a(n) = least natural number k such that EulerPhi(n+k)=EulerPhi(n)+EulerPhi(k), if k exists; otherwise 0. |
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+0
4
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| 2, 1, 0, 4, 4, 4, 14, 6, 6, 4, 16, 6, 14, 6, 0, 5, 8, 6, 6, 8, 0, 4, 46, 12, 10, 8, 6, 12, 26, 12, 62, 6, 12, 4, 16, 12, 28, 6, 0, 10, 24, 24, 86, 8, 0, 6, 38, 6, 62, 25, 12, 16, 24, 18, 32, 24, 0, 4, 118, 24, 80, 6, 12, 10, 28, 12, 134, 8, 0, 35, 142, 24, 146, 8, 30, 12, 8, 24, 46, 20, 6
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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It would be nice to remove the word "Conjectured" from the description - njas
The values of a(3), a(15) and a(21) listed above, namely 0, are conjectural. There are is no natural number k < 10^6 satisfying the "homomorphic condition" EulerPhi(n+k)=EulerPhi(n)+EulerPhi(k) for n=3,5,21.
The terms for which there is no solution k < 10^6 are n = 3, 15, 21, 39, 45, 57, 69, 105, 147, 165, 177, 195, 213, 273, 285,..., which satisfy n=3 (mod 6). - T. D. Noe (noe(AT)sspectra.com), Jan 20 2004
All n<2000 and k<10^8 have been tested. Sequence A110172 gives the n for which there is no solution k<10^8. For n=1 (mod 3) or n=2 (mod 3), it appears that the least solution k satisfies k<=2n. For n=0 (mod 3), the least k, if it exists, can be greater than 2n. - T. D. Noe (noe(AT)sspectra.com), Jul 15 2005
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B36.
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MATHEMATICA
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a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], EulerPhi[ 1, n + # ] == EulerPhi[ 1, n ] + EulerPhi[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]
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CROSSREFS
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Cf. A000010.
Cf. A091531 (primes p such that k=2p is the smallest solution to phi(p+k)=phi(p)+phi(k)).
Cf. A110173 (least k such that phi(n)=phi(k)+phi(n-k) for 0<k<n).
Adjacent sequences: A066423 A066424 A066425 this_sequence A066427 A066428 A066429
Sequence in context: A077908 A052922 A109167 this_sequence A100887 A073592 A077929
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KEYWORD
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nonn
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 27 2001
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EXTENSIONS
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More terms from T. D. Noe (noe(AT)sspectra.com), Jan 20 2004
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