A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152

Offset: 1

Author

Matthew Vandermast, Dec 04 2010

Keywords

• nonn

Comments

In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n.

A141586 is a subsequence. Is A110821 a subsequence?

Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551.

a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022

Examples

			9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors).

Cf. A010553 (d(d(n))), A036450 (d^3(n)), A036452 (d^4(n)), A036453 (d^5(n)).

Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).

Programs

PARI

is_A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022

Extensions

Edited by M. F. Hasler, Apr 16 2022