A068997 Numbers k such that Sum_{d|k} d*mu(d) divides k.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648

Offset: 1

Author

Benoit Cloitre, Apr 07 2002

Keywords

• easy
• nonn

Comments

Numbers k such that A023900(k) divides k.

The only squarefree terms so far are a(1), a(2), and a(4). - Torlach Rush, Dec 04 2017

There are no more squarefree terms. The squarefree terms are also the squarefree terms of A007694 since A023900(n) = A008683(n) * A000010(n) for squarefree numbers n, and A007694 contains only 3-smooth numbers (A003586). - Amiram Eldar, Apr 19 2025

There is a surjective mapping from all even numbers not in this sequence to terms of the sequence. The first such is 10 to a(9). The next is 14, 28, 42 to a(19). All even numbers not in the sequence are divisors of some term in the sequence. - Torlach Rush, Dec 08 2017

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A000010, A003586, A007694, A008683, A023900, A173557, A293928.

Programs

Haskell

a068997 n = a068997_list !! (n - 1)
a068997_list = filter (\x -> mod x (a173557 x) == 0) [1..]
-- Reinhard Zumkeller, Jun 01 2015

Maple

with(numtheory): A068997 := i->`if`(i mod phi(mul(j,j=factorset(i)))=0,i,NULL): seq(A068997(i),i=1..650); # Peter Luschny, Nov 02 2010

Mathematica

Select[Range[650], Divisible[#, DivisorSum[#, # MoebiusMu[#] &]] &] (* Michael De Vlieger, Nov 20 2017 *)
q[1] =True; q[n_] := Divisible[n, Times @@ ((First[#] - 1) & /@ FactorInteger[n])]; Select[Range[650], q] (* Amiram Eldar, Apr 19 2025 *)

PARI

for(n=1,1000,if(n%sumdiv(n,d,moebius(d)*d)==0,print1(n,",")))

PARI

isok(k) = !(k % vecprod(apply(x -> 1-x, factor(k)[, 1]))); \\ Amiram Eldar, Apr 19 2025