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

%I #38 May 09 2023 22:29:07

%S 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,

%T 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,

%U 1,1,2,1,3,4,1,3,2,1,3,1,1,2,2,1,3,3,1,1

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

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

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

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

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

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

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

%H Christian N. K. Anderson, <a href="/A362681/b362681.txt">Table of n, a(n) for n = 1..10000</a>

%H Christian N. K. Anderson, <a href="/A362681/a362681.pdf">Graph showing sparsity of 6s</a>

%o (PARI) a(n) = my(ret=0); while(n>2, n = 4*n+1-sigma(2*n); ret++); ret; \\ _Kevin Ryde_, May 09 2023

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

%K nonn,easy

%O 1,5

%A _Kevin L. Schwartz_ and _Christian N. K. Anderson_, May 01 2023