A388722 a(n) is the number of steps to reach 0 or the start of a cycle using the iteration x -> |sigma(x)-2*x| starting from x=n.

0, 1, 2, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 3, 5, 3, 3, 1, 2, 3, 4, 1, 4, 3, 3, 5, 6, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 2, 5, 3, 4, 2, 3, 3, 3, 2, 4, 4, 5, 3, 4, 2, 3, 1, 4, 3, 4, 3, 4, 2, 3, 5, 6, 4, 4, 3, 5, 3, 4, 4, 5, 3, 4, 3, 3, 4, 4, 2, 3, 4, 3, 2, 5

Offset

1,3

Comments

Every number's next step in its path is abs(A033879) or abs(A033880), which is its aliquot difference.

The only 2-cycle loop known is between 45840 and 51168. Since they are in a loop already and take 0 steps to reach the loop, a(45840)=a(51168)=0. Higher loop sizes exist as well in A371921.

Links

Table of n, a(n) for n=1..93.

Formula

a(p) = a(p-1) + 1.

a(2^n) = 1.

a(perfect) = 1.

a(2p) = a(p-3) + 1.

a(triperfect) = 0.

Example

a(9)=3: iteration 9 -> 5 -> 4 -> 1 (-> 1).
a(60)=3: iteration 60 -> 48 -> 28 -> 0.
a(45840)=0: iteration 45840 (-> 51168 -> 45840).

Mathematica

a[0] = a[1] = 0; a[n_] := a[n] = 1 + a[Abs[DivisorSigma[1, n] - 2*n]]; Array[a, 100] (* Amiram Eldar, Jan 29 2026 *)

Prog

(PARI) a(n) = my(c=0, M=Map()); while(n && !mapisdefined(M, n), mapput(M, n, c); c++; n=abs(sigma(n)-2*n)); if(n, mapget(M, n), c) \\ Andrew Howroyd, Jan 20 2026

Crossrefs

Cf. A033879, A033880.

Cf. A068403, A371921.

Cf. A098007, A098008.

Cf. A113285.

Cf. A000396, A005820.

Sequence in context: A176725 A085029 A376798 * A185318 A008622 A029414

Adjacent sequences: A388719 A388720 A388721 * A388723 A388724 A388725

Keyword

nonn

Author

Miles Reed, Jan 13 2026

Status

approved