A174457 - OEIS

%I #34 Apr 21 2022 13:49:56

%S 1,2,12,24,36,60,72,84,96,108,132,156,180,204,228,240,252,276,288,348,

%T 360,372,396,444,468,480,492,504,516,564,600,612,636,640,672,684,708,

%U 720,732,792,804,828,852,864,876,936,948,972,996,1044,1056,1068,1116,1152

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

%C 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.

%C A141586 is a subsequence. Is A110821 a subsequence?

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

%C 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

%H Amiram Eldar, <a href="/A174457/b174457.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 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.

%o (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

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

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

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

%K nonn

%O 1,2

%A _Matthew Vandermast_, Dec 04 2010

%E Edited by _M. F. Hasler_, Apr 16 2022