A110179 Least k such that phi(n+k) = 2*phi(n), where phi is Euler's totient function.

2, 1, 2, 1, 10, 2, 6, 7, 4, 5, 14, 3, 22, 7, 2, 1, 34, 3, 18, 12, 14, 3, 46, 8, 16, 9, 10, 7, 58, 2, 30, 19, 8, 17, 30, 3, 36, 19, 26, 11, 82, 3, 86, 11, 20, 23, 94, 3, 80, 5, 34, 13, 106, 3, 68, 9, 16, 29, 118, 4, 82, 15, 10, 21, 32, 9, 94, 17, 20, 34, 142, 32, 112, 17, 48, 15, 66, 26

Offset

1,1

Comments

Makowski shows that a k exists for each n. It appears that k <= 2n. For prime n, it appears that n-1 <= k <= 2n.

References

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B36, p. 138.

Links

Donovan Johnson, Table of n, a(n) for n = 1..10000

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Andrzej Makowski, On the equation phi(n+k)=2*phi(n), Elem. Math., Vol. 29, No. 1 (1974), p. 13.

Mathematica

Table[k=1; e=EulerPhi[n]; While[EulerPhi[n+k] != 2e, k++ ]; k, {n, 100}]

Prog

(PARI) a(n) = vecmin(select(x -> x > n, invphi(2*eulerphi(n)))) - n; \\ Amiram Eldar, Nov 05 2024, using Max Alekseyev's invphi.gp

Crossrefs

Cf. A000010, A050473 (least k such that phi(n+k) = 2*phi(k)).

Sequence in context: A062347 A124781 A124151 * A071559 A071560 A248516

Adjacent sequences: A110176 A110177 A110178 * A110180 A110181 A110182

Keyword

nonn

Author

T. D. Noe, Jul 15 2005

Status

approved