A362681 The number of steps, starting from n, to reach x<=2 in an iteration x <- 2x - {sum of proper factors of 2x}.

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

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, May 01 2023

Keywords

• nonn
• easy

Comments

A proper factor is defined as any divisor of n other than 1 and itself (Derbyshire).

The iteration step is x <- A157449(2x).

The iteration ends on the step after reaching half of any abundant number A005101/2.

a(1682)=7 is the only number over 6 in the first 10^6 terms.

Powers of 2 reach 2 in the first step, and then would enter an infinite loop if the iteration ended only when x <= 1.

References

• J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Penguin, 2004, p. 32.

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
• Christian N. K. Anderson, Graph showing sparsity of 6s

Crossrefs

Cf. A157449, A005101, A362684 (indices of records).

Programs

PARI

a(n) = my(ret=0); while(n>2, n = 4*n+1-sigma(2*n); ret++); ret; \\ Kevin Ryde, May 09 2023