A098174 a(n) is the smallest e > 0 such that the initial digit of n^e = 1 in decimal representation.

1, 4, 9, 2, 3, 4, 5, 8, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 3, 3, 3, 3, 3, 3, 5, 7, 9, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 11, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 13, 14, 16, 18, 20, 23, 27, 32, 40, 53

Offset: 1

Author

Reinhard Zumkeller, Aug 30 2004

Keywords

• nonn
• base

Comments

A000030(n^a(n)) = 1; A098175(n) = n^a(n).

From Rémy Sigrist, Jun 25 2018: (Start)

We can extend this sequence to every Gaussian integers as follows:

- for any Gaussian integer z, let f(z) be the least k > 0 such that the initial decimal digit of the real part of z^k equals 1, or -1 if no such k exists,

- naturally f(n) = a(n) for any n > 0,

- apparently f(z) = -1 iff z = 0,

- see Links section for the color plot of f.

(End)

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Color plot of f(x + i*y) for x = -200..1000 and y = -200..500

Crossrefs

Cf. A000030, A098175, A131835.

Programs

PARI

a(n, base=10) = my (nk=n); for (k=1, oo, my (z); logint(nk, base, &z); if (nk\z==1, return (k), nk*=n)) \\ Rémy Sigrist, Jun 21 2018