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A066426 Conjectured values for a(n) = least natural number k such that phi(n+k) = phi(n) + phi(k), if k exists; otherwise 0.

%I A066426 #18 May 15 2022 17:22:23
%S A066426 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,
%T A066426 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,
%U A066426 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
%N A066426 Conjectured values for a(n) = least natural number k such that phi(n+k) = phi(n) + phi(k), if k exists; otherwise 0.
%C A066426 It would be nice to remove the word "Conjectured" from the description. - _N. J. A. Sloane_
%C A066426 The values of a(3), a(15) and a(21) listed above, namely 0, are conjectural. There is no natural number k < 10^6 satisfying the "homomorphic condition" phi(n+k) = phi(n) + phi(k) for n = 3, 15, 21.
%C A066426 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_, Jan 20 2004
%C A066426 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_, Jul 15 2005
%D A066426 R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B36.
%t A066426 a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], EulerPhi[ 1, n + # ] == EulerPhi[ 1, n ] + EulerPhi[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]
%Y A066426 Cf. A000010.
%Y A066426 Cf. A091531 (primes p such that k=2p is the smallest solution to phi(p+k) = phi(p) + phi(k)).
%Y A066426 Cf. A110173 (least k such that phi(n) = phi(k) + phi(n-k) for 0 < k < n).
%K A066426 nonn
%O A066426 1,1
%A A066426 _Joseph L. Pe_, Dec 27 2001
%E A066426 More terms from _T. D. Noe_, Jan 20 2004