A393869 Numbers k such that A053669(k) < A003557(k).

8, 9, 16, 25, 27, 32, 36, 40, 45, 48, 49, 50, 54, 56, 63, 64, 72, 75, 80, 81, 88, 96, 98, 99, 100, 104, 108, 112, 117, 121, 125, 128, 135, 136, 144, 147, 152, 153, 160, 162, 169, 171, 175, 176, 184, 189, 192, 196, 200, 207, 208, 216, 224, 225, 232, 240, 242, 243

Offset

1,1

Comments

Let q = A053669(k) be the smallest prime such that gcd(k,q) = 1. Let squarefree kernel rad = A007947 and let m = A003557(k) = k/rad(k). Then k is in this sequence if q < m.

Proper subset of A013929 (nonsquarefree numbers); does not intersect A005117.

Union of A246547 \ {4} and A360765.

{a(n)} \ A001694 = {1, 4}.

Agrees with A388986 for 55 terms; a(56) = 240 is not in A388986.

Terms are such that there exists a number j < k such that rad(k) | gcd(j,k). This is to say that, though rad(k) | j, j has a factor greater than 1 that does not divide k.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Diagram showing j <= k, n = 1..72, where a number k is a term of this sequence (written at right in red) for which there exists at least one j <= k (highlighted in blue, which appear in row k of A272619) such that gcd(j,k) > 1, rad(j) does not divide k, but rad(k) | j. Green dots also appear in row k of A272619 but in those cases, rad(k) does not divide j.

Michael De Vlieger, Plot j < k at (x,y) = (j,k), k = 1..1024, such that gcd(j,k) > 1, rad(j) does not divide k, and rad(k) | j. In the plot, j increases toward the right and k increases downward.

Example

Let r = rad(a(n)) and m = A381803(n).
Table of n, a(n), r, and j for select n:
     n    a(n)   r   j
  ----------------------------------------------------------------------------
     1      8    2  {6}
     2      9    3  {6}
     3     16    2  {6, 10, 12, 14}
     4     25    5  {10, 15, 20}
     5     27    3  {6, 12, 15, 18, 21, 24}
     6     32    2  {6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 30}
     7     36    6  {30}
     8     40   10  {30}
    17     72    6  {30, 42, 60, 66}
    35    144    6  {30, 42, 60, 66, ..., 138}
    56    240   30  {210}
  1331   5184    6  {30, 42, 60, 66, ..., 5178}

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; a053669[x_] := Module[{qx}, qx = 2; While[Divisible[x, qx], qx = NextPrime[qx]]; qx]; Select[Range[250], a053669[#] < #/rad[#] &]

Crossrefs

Cf. A001694, A003557, A005117, A007947, A013929, A053669, A246547, A388986.

Proper subsets: A052486, A286708, A303606, A360765, A361098, A366250, A366460, A367708, A369150, A383394, A386762, A389864.

Sequence in context: A346090 A169817 A351098 * A388986 A396324 A227649

Adjacent sequences: A393866 A393867 A393868 * A393870 A393871 A393872

Keyword

nonn,easy

Author

Michael De Vlieger, Jun 02 2026

Status

approved