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

Author

Joseph L. Pe, Dec 27 2001

Keywords

• nonn

Comments

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

References

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

Crossrefs

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).

Programs

Mathematica

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

Extensions

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