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

1,7

Links

T. D. Noe, Table of n, a(n) for n = 1..200

Michael Penn, From the GOAT number theory book!, YouTube video, 2022.

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

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 *)

Prog

(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

Crossrefs

Cf. A000142, A051626.

Sequence in context: A315818 A315819 A301682 * A315820 A315821 A315822

Adjacent sequences: A000973 A000974 A000975 * A000977 A000978 A000979

Keyword

nonn,base

Author

Simon Plouffe

Extensions

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

Status

approved