A300234 a(n) = number of steps in simple Euclidean algorithm for gcd(n,k) to reach the termination test n=k when starting with n = n and k = phi(n).

0, 1, 2, 1, 4, 2, 6, 1, 2, 3, 10, 2, 12, 4, 8, 1, 16, 2, 18, 3, 4, 6, 22, 2, 4, 7, 2, 4, 28, 6, 30, 1, 9, 9, 9, 2, 36, 10, 5, 3, 40, 4, 42, 6, 8, 12, 46, 2, 6, 3, 10, 7, 52, 2, 5, 4, 6, 15, 58, 6, 60, 16, 4, 1, 10, 8, 66, 9, 11, 13, 70, 2, 72, 19, 8, 10, 13, 6, 78, 3, 2, 21, 82, 4, 24, 22, 12, 6, 88, 6, 11, 12, 8, 24, 13, 2, 96, 4, 9, 3, 100, 10, 102, 7, 9

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Antti Karttunen, Scheme (Racket) program to compute this sequence

Formula

a(n) = A285721(n,A000010(n)).

a(n) = n - A300238(n).

Example

For n = 1, phi(1) = 1, and the arguments for gcd are equal at the start, thus a(1) = 0.
For n = 2, eulerphi(2) = 1, gcd(2,1) = gcd(1,1), thus 1 step were required to reach the termination condition, and a(2) = 1.
For n = 5, eulerphi(5) = 4, gcd(5,4) = gcd(4,1) = gcd(3,1) = gcd(2,1) = gcd(1,1), four steps required, thus a(5) = 4.
For n = 6, eulerphi(6) = 2, gcd(6,2) = gcd(4,2) = gcd(2,2), two steps required, thus a(6) = 2.
Here a simple subtracting version of gcd-algorithm is used, where the new versions of two arguments will be the smaller argument and the smaller argument subtracted from the larger, and this is repeated until both are equal.

Prog

(PARI)
A285721(n, k) = if(n==k, 0, 1 + A285721(abs(n-k), min(n, k)));
A300234(n) = A285721(n, eulerphi(n));

Crossrefs

Cf. A000010, A285721.

Cf. also A286594, A300227, A300228, A300237, A300238.

Sequence in context: A147763 A342415 A098371 * A070777 A173614 A173557

Adjacent sequences: A300231 A300232 A300233 * A300235 A300236 A300237

Keyword

nonn

Author

Antti Karttunen, Mar 02 2018

Status

approved