A157449 -id:A157449 - cp's OEIS frontend

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

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

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.

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

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

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

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

A362684 a(n) is the index at which n first occurs in A362681.

Original entry on oeis.org

1, 3, 5, 7, 26, 49, 632, 1682

Offset: 0

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

Among terms n=1..7, each number iterates through the previous terms. For example, the first iteration takes 1682 to 632=a(6), the second takes 632 to 49=a(5) and so forth.
1682 is the only number < 10^6 that requires 7 iterations to reach completion.
a(8), if it exists, is larger than 10^6.
a(8) > 10^9. - Michel Marcus, Oct 11 2023

			a(7) = 1682 because the first number to require 7 iterations to reach 2 or less is 1682.
Each iteration step is n <- 2n - {sum of proper factors of 2n} = A157449(2n); for n = 1682, this gives 2n = 3364, 3364 - (2 + 4 + 29 + 58 + 116 + 841 + 1682) = 632, which is a(6).
Note that the proper factors of x are all divisors of x other than 1 and x.

Cf. A362681, A157449.

Cf. A048133.

a(0)=1 prepended by Michel Marcus, Oct 11 2023

A180268 Numbers n, not divisible by 3, 5, 7 or 11, such that A000203(n)-n-1 and 2*n+1-A000203(n) are prime numbers.

Original entry on oeis.org

4, 988, 1612, 3172, 5332, 7852, 11908, 18748, 19276, 21268, 21508, 25156, 32188, 35932, 41116, 41452, 43108, 50236, 54436, 55876, 57412, 62452, 65836, 71248, 72748, 74572, 85828, 88996, 96772, 97708, 99292, 103948, 104956, 105772, 110812, 116356, 116428

Offset: 1

Alexey Galkin (fetch91(AT)gmail.com), Aug 23 2010

Cf. A157449, A175727 and A179945.

Divisibility requirement added to definition to match the sequence; 4 and terms beyond 7852 added - R. J. Mathar, Aug 25 2010

cp's OEIS Frontend

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

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A362684 a(n) is the index at which n first occurs in A362681.

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A180268 Numbers n, not divisible by 3, 5, 7 or 11, such that A000203(n)-n-1 and 2*n+1-A000203(n) are prime numbers.

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