A262981 Numbers k such that the least positive integer having exactly k divisors is divisible by k.

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

Author

Vladimir Letsko, Oct 06 2015

Keywords

• nonn

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)

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Anton Nikonov)
• Vladimir Letsko, The table of correspondence between A262981 and A262983
• Vladimir Letsko, Mathematical Marathon, Problem 216 (in Russian)

Crossrefs

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

Programs

PARI

fhasndiv(n) = {k=1; while (numdiv(k) != n, k++); k;}
isok(n) = if (!(fhasndiv(n) % n), 1, 0); \\ Michel Marcus, Oct 06 2015

Extensions

Missing a(34) added by Giovanni Resta, Oct 06 2015