A396265 Numbers k such that there exists a number r < k such that both gcd(r,k) > 1 and rad(r) does not divide k, yet rad(r+m*k) | k for some m, where rad = A007947.

10, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset

1,1

Comments

Numbers k such that there exists a number r in row k of A272619 (i.e., [1..k] \ union({row k of A038566}, {row k of A162306})) such that rad(r+m*k) | k for some m >= 0.

Powers of primes p^m, m >= 0 (p^m in A000961) are such that row p^m of A272619 and row p^m of A381801 do not intersect.

This sequence is A024619 \ A382438.

Conjecture: includes A126706 \ {12, 24}.

Links

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

Example

Let s = r + m * k.
6 is not in this sequence since r < 6 is either coprime to 6 or rad(r) | 6.
10 is in this sequence since for r = 6 and s = 16 = 6+1*10, rad(16) | 10, while gcd(6,10)=2 and rad(6) does not divide 10.
12 is not in this sequence since row 12 of A038566 is {1, 5, 7, 11} and row 12 of A162306 is {1, 2, 3, 4, 6, 8, 9}, leaving 10 in row 12 of A272619. However, numbers congruent to 10 (mod 12) are not divisible by 3. Therefore there does not exist a number r such that rad(r+12*m) | 12 for some m >= 0.
14 is not in this sequence since row 14 of A272619 is {6, 10, 12}, and the prime power residues of 2 (mod 14) are {1,2,4,8} and of 7 are {1,7}, thus row 14 of A381801 is {0, 1, 2, 4, 7, 8}; these sets are disjoint. Therefore there does not exist a number r such that rad(r+12*m) | 12 for some m >= 0.
15 is in this sequence since row 15 of A272619 = {6, 10, 12} and row 15 of A381801 = {0, 1, 3, 5, 6, 9, 10, 12} intersect, etc.

Mathematica

nn = 120; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
a382438[x_] := Module[{p}, Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, x] &, 1, UnsameQ, All]) &, FactorInteger[x][[All, 1]] ], x] ];
a024619 = Select[Range[2, nn], Not @* PrimePowerQ];
Select[a024619, Function[k, ! AllTrue[Rest@ a382438[k], Divisible[k, rad[#] ] &] ] ]

Crossrefs

Cf. A000961, A007947, A024619, A038566, A126706, A162306, A272619, A381800, A381801, A381804, A382438.

Sequence in context: A269985 A282648 A139540 * A396948 A343550 A285176

Adjacent sequences: A396262 A396263 A396264 * A396266 A396267 A396268

Keyword

nonn

Author

Michael De Vlieger, May 21 2026

Status

approved