A242028 Numbers k such that the least common multiple of the anti-divisors of k is less than k.

3, 4, 6, 9, 36, 54, 96, 216, 576, 1296, 69984, 236196, 393216, 497664, 28697814, 31850496, 84934656, 2038431744, 5435817984, 1174136684544, 28179280429056, 39582418599936, 42268920643584, 75144747810816, 289207845356544, 570630428688384

Offset

1,1

Comments

Numbers k such that lcm(anti-divisors(k)) < lcm(divisors(k)).

Each term k has form 2^s or 2^s*p^t with an odd prime p as otherwise LCM of the even anti-divisors of k equals 2k > k. The case 2^s implies that 2k-1 or 2k+1 is a power of 3, yielding only term a(2) = 4 per Mihailescu's theorem. LCM of the even anti-divisors of k = 2^s*p^t equals 2^(s+1)*p^(t-1), further implying that both 2k-1 and 2k+1 must be prime. Finally, it follows that p = 3, ie. all terms have form k = 2^s*3^t with both 2k-1 and 2k+1 being prime, except for k = 4. - Max Alekseyev, Sep 16 2025

Links

Table of n, a(n) for n=1..26.

Formula

For n >= 3, a(n) = A027856(n)/2. - Max Alekseyev, Sep 16 2025

Example

a(6) = 9 because lcm(2, 6) = 6, which is less than 9.

Mathematica

antiDivisors[n_Integer] :=
  Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)];
a242028[n_Integer] := Select[Range[n],
  Length[antiDivisors[#]] > 0 && LCM @@ antiDivisors[#] < # &]; a242028[5000] (* Michael De Vlieger, Aug 21 2014 *)

Prog

(Python)
from sympy import divisors, lcm
A242028 = [n for n in range(3, 10**5) if lcm(
          [2*d for d in divisors(n) if n > 2*d and n % (2*d)] +
          [d for d in divisors(2*n-1) if n > d >=2 and n % d] +
          [d for d in divisors(2*n+1) if n > d >=2 and n % d]) < n]
# Chai Wah Wu, Aug 19 2014

Crossrefs

Cf. A027856, A096357.

Sequence in context: A191699 A293272 A192286 * A254002 A095729 A185739

Adjacent sequences: A242025 A242026 A242027 * A242029 A242030 A242031

Keyword

nonn

Author

Michael De Vlieger, suggested by Michel Marcus, Aug 11 2014

Extensions

Corrected comment, removed 1 and 2 from sequence and added a(11)-a(16) by Chai Wah Wu, Aug 20 2014

Terms a(17) onward added by Max Alekseyev, Sep 16 2025

Status

approved