A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 84, 88, 92, 98, 99, 104, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 184, 188, 189, 198, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 242, 244

Offset

1,1

Links

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

Formula

Intersection of A332785 and A080259 = A332785 \ A055932 = A126706 \ A286708 \ A380543.

Example

Table of n, a(n) and prime decomposition for n = 1..12:
 n  a(n)
------------------
 1   20 = 2^2 * 5
 2   28 = 2^2 * 7
 3   40 = 2^3 * 5
 4   44 = 2^2 * 11
 5   45 = 3^2 * 5
 6   50 = 2 * 5^2
 7   52 = 2^2 * 13
 8   56 = 2^3 * 7
 9   63 = 3^2 * 7
10   68 = 2^2 * 17
11   75 = 3 * 5^2
12   76 = 2^2 * 19
Let q = A053669 and let gpf = A006530.
The number 12 = 2^2*3 is not in the sequence since q(12) > gpf(12), i.e., 5 > 3.
The number 18 = 2*3^2 is not in the sequence since q(18) > gpf(18), i.e., 5 > 3.
a(1) = 20 = 2^2*5 since q(20) < gpf(20), i.e., 3 < 5.
The number 60 = 2^2*3*5 is not a term since q(60) > gpf(60), i.e., 7 > 5, etc.

Mathematica

f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q]; Select[Range[256], Nor[Length[#2] == 1, Max[#2[[All, -1]]] == 1, Divisible[#1, Apply[Times, #2[[All, 1]]]^2], f[#1] > #2[[-1, 1]]] & @@ {#, FactorInteger[#]} &]

Crossrefs

Cf. A001694, A002110, A006939, A007947, A013929, A052485, A053669, A055932, A080259, A126706, A286708, A332785, A380543.

Sequence in context: A367632 A137428 A368089 * A367455 A350841 A361867

Adjacent sequences: A386291 A386292 A386293 * A386295 A386296 A386297

Keyword

nonn

Author

Michael De Vlieger, Jul 19 2025

Status

approved