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A262981
Numbers k such that the least positive integer having exactly k divisors is divisible by k.
4
1, 2, 6, 8, 9, 12, 18, 20, 24, 30, 36, 40, 45, 56, 60, 72, 75, 80, 84, 90, 112, 120, 125, 126, 140, 144, 150, 168, 180, 210, 224, 225, 240, 250, 252, 264, 280, 288, 300, 315, 336, 350, 352, 360, 375, 396, 420, 440, 441, 448, 450, 500, 504, 525, 528, 560, 600, 616, 624, 625
OFFSET
1,2
COMMENTS
The sequence contains numbers n for which A005179(n) is a multiple of n.
In turn, A002110 is a subsequence.
From David A. Corneth, Aug 21 2016: (Start)
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.
Elements > 1 have the property primepi(p) <= bigomega(n). For 2 <= k <= 100000, only 2114 values k have this property. (End)
From Vladimir Letsko, Dec 11 2016: (Start)
The first comment in other words: a positive integer n is in this sequence iff A005179(n) is in A033950.
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.
Let p be an odd prime and s > 0. Then p^s is in the sequence if and only if pi(p) <= s < p.
Let k > 1. There are infinitely many k such that n^k is in the sequence.
Some conjectures for a(n):
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).
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)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Anton Nikonov)
Vladimir Letsko, Mathematical Marathon, Problem 216 (in Russian)
EXAMPLE
9 is in the sequence because the least positive integer having exactly 9 divisors is 36, which is divisible by 9.
PROG
(PARI) fhasndiv(n) = {k=1; while (numdiv(k) != n, k++); k; }
isok(n) = if (!(fhasndiv(n) % n), 1, 0); \\ Michel Marcus, Oct 06 2015
KEYWORD
nonn
AUTHOR
Vladimir Letsko, Oct 06 2015
EXTENSIONS
Missing a(34) added by Giovanni Resta, Oct 06 2015
STATUS
approved