A091531 -id:A091531 - cp's OEIS frontend

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

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

Offset: 1

Joseph L. Pe, Dec 27 2001

nonn

It would be nice to remove the word "Conjectured" from the description. - N. J. A. Sloane
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.
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
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

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B36.

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).

a[ n_ ] := Min[ Select[ Range[ 1, 10^6 ], EulerPhi[ 1, n + # ] == EulerPhi[ 1, n ] + EulerPhi[ 1, # ] & ] ]; Table[ a[ i ], {i, 1, 21} ]

More terms from T. D. Noe, Jan 20 2004

cp's OEIS Frontend

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

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