A279373 -id:A279373 - cp's OEIS frontend

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

Vladimir Letsko, Oct 06 2015

nonn

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)

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

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)

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

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

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

A262983 Terms of A005179 divisible by their indices in order of appearance in A005179.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3600, 6720, 5040, 10080, 32400, 15120, 20160, 25200, 60480, 55440, 810000, 100800, 181440, 110880, 226800, 221760, 277200, 907200, 665280, 1587600, 720720, 5670000, 1108800, 3548160, 1995840, 1441440, 2494800, 6350400

Offset: 1

Vladimir Letsko, Oct 06 2015

A005179(n) is in this sequence iff it is divisible by n. Thus this is a subsequence of A005179 indexed by A262981.
Also this sequence is the intersection of A033950 and A005179. Hence this sequence has density zero. - Vladimir Letsko, Dec 16 2016
It seems that this sequence is a subsequence of A262981.
This sequence is not in ascending order as terms of A005179 divisible by their number of divisors do not occur in ascending order. For terms sorted in ascending order see A110821. - David A. Corneth, Dec 10 2021

			12 is a term since it is the smallest positive integer having exactly 6 divisors and divisible by 6.

David A. Corneth, Table of n, a(n) for n = 1..10000
Vladimir Letsko, The table of correspondence between A262983 and A262981

Cf. A000005, A033950, A005179, A110821, A262981, A279373.

Take[#, 33] &@ DeleteCases[#, 0] &@ Function[s, ReplacePart[#, Flatten@ Map[{# -> Function[k, k Boole[Divisible[k, #]]]@ Lookup[s, #]} &, Keys@ s]] &@ ConstantArray[0, Max@ Keys@ s]]@ Map[First, KeySort@ PositionIndex@ Table[DivisorSigma[0, n], {n, 10^7}]] (* Michael De Vlieger, Dec 11 2016, Version 10 *)

a(n) = A005179(A262981(n)).

A000005(a(n)) = A262981(n).

Name clarified by David A. Corneth, Dec 10 2021

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

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