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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. 3
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; refs; listen; history; text; internal format)
OFFSET

1,1

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, 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, 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.

LINKS

Table of n, a(n) for n=1..81.

MATHEMATICA

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

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).

Sequence in context: A181930 A256797 A109167 * A100887 A073592 A164994

Adjacent sequences:  A066423 A066424 A066425 * A066427 A066428 A066429

KEYWORD

nonn

AUTHOR

Joseph L. Pe, Dec 27 2001

EXTENSIONS

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

STATUS

approved

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Last modified April 12 16:24 EDT 2021. Contains 342927 sequences. (Running on oeis4.)