A034883 Maximum length of Euclidean algorithm starting with n and any nonnegative i

0, 1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 4, 5, 4, 4, 4, 4, 5, 5, 4, 6, 4, 5, 4, 5, 5, 5, 5, 6, 6, 6, 5, 5, 7, 5, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 7, 7, 6, 6, 6, 5, 8, 6, 6, 6, 6, 7, 6, 6, 6, 7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 8, 6, 6, 8, 8, 8, 7, 7, 6, 7, 7, 7, 7, 9, 6, 7, 7, 7, 7, 7, 6, 8, 7, 7, 7

Offset: 1

Author

Erich Friedman

Keywords

• easy
• nonn

Comments

Apart from initial term, same as A071647. - Franklin T. Adams-Watters, Nov 14 2006

Records occur when n is a Fibonacci number. For n>1, the smallest i such that the algorithm requires a(n) steps is A084242(n). The maximum number of steps a(n) is greater than k for n > A188224(k). - T. D. Noe, Mar 24 2011

Largest term in n-th row of A051010. - Reinhard Zumkeller, Jun 27 2013

a(n)+1 is the length of the longest possible continued fraction expansion (in standard form) of any rational number with denominator n. - Ely Golden, May 18 2020

Links

• T. D. Noe, Table of n, a(n) for n=1..1000
• Eric Weisstein's World of Mathematics, Euclidean Algorithm

Programs

Haskell

a034883 = maximum . a051010_row  -- Reinhard Zumkeller, Jun 27 2013

Mathematica

GCDSteps[n1_, n2_] := Module[{a = n1, b = n2, cnt = 0}, While[b > 0, cnt++; {a, b} = {Min[a, b], Mod[Max[a, b], Min[a, b]]}]; cnt]; Table[Max @@ Table[GCDSteps[n, i], {i, 0, n - 1}], {n, 100}] (* T. D. Noe, Mar 24 2011 *)

Python

def euclid_steps(a,b):
    step_count = 0
    while(b != 0):
        a , b = b , a % b
        step_count += 1
    return step_count
for n in range(1,1001):
    l = 0
    for i in range(n): l = max(l,euclid_steps(n,i))
    print(str(n)+" "+str(l)) # Ely Golden, May 18 2020