A185358 The period of the sequence i^i (mod n) starts from i=a(n).

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1

Offset

1,8

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

R. Hampel, The length of the shortest period of rests of numbers n^n, Ann. Polon. Math. 1 (1955), 360-366.

Formula

If n = Product_{pi^ei} then a(n) = Max_{1- pi*(1+floor[-ei/pi])}.

Mathematica

a[p_, e_]:=1- p*(1+Floor[-e/p]); a[n_]:=Max@Module[{fa=FactorInteger[n]}, Table[a[fa[[i, 1]], fa[[i, 2]]], {i, 1, Length[fa]}]]; Table[a[n], {n, 1, 84}]

Prog

(Python)
from sympy import factorint, floor
def a(n):
    f=factorint(n)
    return 1 if n==1 else max(1 - i*(1 + (-f[i])//i) for i in f)
print([a(n) for n in range(1, 201)]) # Indranil Ghosh, Jun 29 2017

Crossrefs

Cf. A185359.

Sequence in context: A031229 A031228 A031227 * A368247 A372604 A331273

Adjacent sequences: A185355 A185356 A185357 * A185359 A185360 A185361

Keyword

nonn

Author

José María Grau Ribas, Jan 21 2012

Status

approved