A396324 Numbers k for which there exists a residue class r (mod k) with s = r+m*k, m >= 0, such that rad(s) | gcd(s,k), where rad = A007947.

8, 9, 16, 25, 27, 32, 36, 40, 48, 49, 63, 64, 72, 80, 81, 96, 98, 100, 104, 108, 112, 117, 121, 125, 128, 136, 144, 160, 169, 171, 176, 189, 192, 196, 200, 208, 216, 224, 225, 232, 243, 252, 256, 272, 275, 279, 288, 289, 296, 304, 320, 324, 328, 333, 343, 351

Offset

1,1

Comments

Squarefree k are such that rad(k) | s implies k | s, thus do not appear in this sequence.

Superset of A246547 / {4} (proper prime powers), superset of A303606 (perfect powers of squarefree composites), thus, superset of their union, A072777 / {4}.

Intersects A059404 and its proper subsets A332785 (e.g., a(9) = 40), A052486 (Achilles, e.g., a(14) = 72), A386762 (e.g., a(28) = 144), and A383394 (e.g., a(702) = 5184).

Conjecture: A359280 is a proper subset (and thus, all powerful numbers greater than 4, i.e., A001694 \ {1, 4} = A359280 U A072777 / {4}).

Does not meet A366825, since for k in A366825, k = lpf(k)*rad(k). Therefore, since lpf(k)*rad(k) is the smallest number that exceeds rad(k), then all r < k such that rad(k) | r (i.e., rad(k) itself) also divide k.

Proper subset of A393869 (numbers k such that A053669(k) < A003557(k)).

Links

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

Example

Let r = rad(a(n)) and m = A381803(n).
Table of n, a(n), r, m, and k for select n:
    n    a(n)    r    m   k
  ----------------------------------------------------
    1      8     2    1   {6}
    2      9     3    1   {6}
    3     16     2    4   {6, 10, 12, 14}
    4     25     5    3   {10, 15, 20}
    7     36     6    1   {30}
    8     40    10    1   {30}
   13     72     6    4   {30, 42, 60, 66}
   18    100    10    3   {30, 70, 90}
   27    144     6   13   {30, 42, 60, 66, 78, 84, 90, 102, 114, 120, 126, 132, 138}
   42    252    42    1   {210}
  457   3276   546    1   {2730}
  701   5184     6  805   {30, 42, 60, 66, ..., 5178}
The number 4 is not a term since 1 and 3 are coprime to 4, and 2 | 4.
a(1) = 8 since r = 6 is such that s = 6 + 8*m, m >= 0, is even, yet not divisible by 8.
a(2) = 9 since r = 6 is such that s = 6 + 9*m, m >= 0, is divisible by 3 but not by 9.
The number k = 12 is not in the sequence since, though there exists r = 10 with s = 10 + 12*m such that rad(s) does not divide 12, gcd(s,12) = 2; s is not divisible by 3.
a(3) = 16 since r in {6, 10, 12, 14} are such that s = r + 16*m, m >= 0, are even but not divisible by 16.
The number 36 is a term, since there exists one residue class r = 30 with s = 30 + 36*m, m >= 0, such that both rad(36) | gcd(s,36) and rad(s) does not divide 36. Examples: 66 = 2*3*11, 112 = 2*3*17, 138 = 2*3*23, etc.
a(9) = 48 since there exists one residue class r in {30 = 6*5, 42 = 6*7} with s = 30 + 48*m such that both rad(36) | gcd(s,36) and rad(s) does not divide 36. Examples: 66 = 2*3*11, 112 = 2*3*17, 138 = 2*3*23, etc.
The number 45 is not a term since, though for r = 30, 1 < gcd(30,45) < 30, s = 30 + 45 = 75 is divisible by rad(45) = 15.

Mathematica

Select[
  Select[Range[360], Not @* SquareFreeQ],
  Function[k, r = rad[k];
    s = Select[Range[k], CoprimeQ[#, k] &];
    t = Union@ Flatten@ Mod[TensorProduct @@
      Map[(p = #; NestWhileList[Mod[p*#, k] &, 1, UnsameQ, All]) &,
       FactorInteger[k][[All, 1]] ], k];
    Count[Complement[Range[0, k - 1], Union[s, t]],
      _?(Divisible[#, r] &)] > 0] ]

Crossrefs

Cf. A003557, A005117, A007947, A013929, A024619, A052486, A053669, A059404, A246547, A303606, A359280, A366825, A381803, A383384, A386762, A393869.

Sequence in context: A351098 A393869 A388986 * A227649 A227648 A192636

Adjacent sequences: A396320 A396322 A396323 * A396325 A396326 A396327

Keyword

nonn,new

Author

Michael De Vlieger, Jun 01 2026

Status

approved