A000976 Period of 1/n! in base 10.

0, 0, 1, 1, 1, 1, 6, 6, 18, 18, 18, 54, 54, 378, 1134, 1134, 9072, 81648, 81648, 81648, 1714608, 18860688, 18860688, 56582064, 56582064, 735566832, 19860304464, 139022131248, 139022131248, 417066393744, 2085331968720, 2085331968720, 68815954967760

Offset: 1

Author

Simon Plouffe

Keywords

• nonn
• base

Links

• T. D. Noe, Table of n, a(n) for n = 1..200
• Michael Penn, From the GOAT number theory book!, YouTube video, 2022.

Crossrefs

Cf. A000142, A051626.

Programs

Mathematica

Join[{0, 0}, Table[num = n!/(2^IntegerExponent[n!, 2] * 5^IntegerExponent[n!, 5]); MultiplicativeOrder[10, num], {n, 3, 30}]] (* T. D. Noe, Jun 21 2012 *)

PARI

a(n) = if(n <= 2, return(0)); znorder(Mod(10,n!/2^val(n,2)/5^val(n,5)))
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Jan 11 2023

Formula

a(n) = k where k is the smallest integer >= 1 such that 10^k == 1 (mod n!/(2^A011371(n)*5^A027868(n))) where A011371(n) is the highest power of 2 dividing n! and A027868(n) is the largest k such that 5^k | n!. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004, corrected by David A. Corneth, Jan 11 2023

a(n) = order(10, n!/(2^s*5^t)) where 2^s is largest power of 2 dividing n! and 5^t is largest power of 5 dividing n!. - Sean A. Irvine, Sep 29 2011

a(n) = A051626(A000142(n)). - Michel Marcus, Jan 12 2023

Extensions

One more term from Sean A. Irvine, Sep 28 2011