A066426 - OEIS

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

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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, 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.

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