A262981 - OEIS

%I #70 Oct 10 2024 08:18:12

%S 1,2,6,8,9,12,18,20,24,30,36,40,45,56,60,72,75,80,84,90,112,120,125,

%T 126,140,144,150,168,180,210,224,225,240,250,252,264,280,288,300,315,

%U 336,350,352,360,375,396,420,440,441,448,450,500,504,525,528,560,600,616,624,625

%N Numbers k such that the least positive integer having exactly k divisors is divisible by k.

%C The sequence contains numbers n for which A005179(n) is a multiple of n.

%C In turn, A002110 is a subsequence.

%C From _David A. Corneth_, Aug 21 2016: (Start)

%C 2 is the only prime in the sequence. Let p be the largest prime divisor of n. If n is in the sequence, then is it true that n/p is in the sequence? Not for n = 20.

%C Elements > 1 have the property primepi(p) <= bigomega(n). For 2 <= k <= 100000, only 2114 values k have this property. (End)

%C From _Vladimir Letsko_, Dec 11 2016: (Start)

%C The first comment in other words: a positive integer n is in this sequence iff A005179(n) is in A033950.

%C Note that p! is in the sequence for all primes p. On the other hand, each number in the run from (2^n)! to q-1, where n>2 and q is the least prime greater than (2^n)!, isn't in the sequence.

%C Let p be an odd prime and s > 0. Then p^s is in the sequence if and only if pi(p) <= s < p.

%C Let k > 1. There are infinitely many k such that n^k is in the sequence.

%C Some conjectures for a(n):

%C 1. Let b be in a(n). Then A005179(b) is in a(n) too. In other words, A262983 is a subsequence of a(n).

%C 2. Let b be any positive integer and b_1 denote A005179(b), b_2 denote A005179(b_1), and so on. Then b_k is in a(n) for some k. (End)

%H Amiram Eldar, <a href="/A262981/b262981.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Anton Nikonov)

%H Vladimir Letsko, <a href="/A262981/a262981.txt">The table of correspondence between A262981 and A262983</a>

%H Vladimir Letsko, <a href="http://www-old.fizmat.vspu.ru/doku.php?id=marathon:problem_216">Mathematical Marathon, Problem 216</a> (in Russian)

%e 9 is in the sequence because the least positive integer having exactly 9 divisors is 36, which is divisible by 9.

%o (PARI) fhasndiv(n) = {k=1; while (numdiv(k) != n, k++); k;}

%o isok(n) = if (!(fhasndiv(n) % n), 1, 0); \\ _Michel Marcus_, Oct 06 2015

%Y Cf. A000005, A000720, A001222, A002110, A002182, A005179, A033950, A037019, A072066, A262983, A279373.

%K nonn

%O 1,2

%A _Vladimir Letsko_, Oct 06 2015

%E Missing a(34) added by _Giovanni Resta_, Oct 06 2015