login
A174457
Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.
2
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
1,2
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
LINKS
EXAMPLE
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.
PROG
(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
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).
Sequence in context: A141079 A269841 A144551 * A353011 A110821 A262983
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Dec 04 2010
EXTENSIONS
Edited by M. F. Hasler, Apr 16 2022
STATUS
approved