A363082 Numbers k neither squarefree nor prime power such that q*r > k, where q = A053669(k) is the smallest prime that does not divide k and r = A007947(k) is the squarefree kernel.

12, 18, 20, 24, 28, 44, 52, 60, 68, 76, 84, 90, 92, 116, 120, 124, 126, 132, 140, 148, 150, 156, 164, 168, 172, 180, 188, 198, 204, 212, 220, 228, 234, 236, 244, 260, 264, 268, 276, 284, 292, 306, 308, 312, 316, 332, 340, 342, 348, 356, 364, 372, 380, 388, 404, 408, 412, 414, 420

Offset

1,1

Comments

All terms are even. Conjectured to be a subsequence of A389216. - Antti Karttunen, Sep 30 2025

Links

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

Michael De Vlieger, Plot b(n) = A126706(n) at (x, y) for n = ym + x = 1..1032256, m = 1016 and x = 1..m, y = 0..m-1, showing b(n) in A360765 in white, and b(n) in this sequence in other colors, where red indicates b(n) also in A360767, and blue indicates b(n) also in A360768.

Formula

This sequence is A126706 \ A360765.

Example

a(1) = 12 since 12 is the smallest number that is neither squarefree nor a prime power. Additionally, 12 < 5*6.
a(2) = 18 since it is in A126706, and like 12, 18 < 5*6.
a(3) = 20 since it is neither squarefree nor prime power, and 20 < 3*10.
36 is not in this sequence since 36 > 5*6.
40 is not in this sequence since 40 > 3*10, etc.

Mathematica

Select[Select[Range[452], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{q, r}, q r > k] @@ {SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]} ] @@ {#, FactorInteger[#]} &]

Prog

(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
A053669(n) = forprime(p=2, , if(n%p, return(p)));
is_A363082(k) = if(issquarefree(k) || isprimepower(k), 0, ((A053669(k)*A007947(k)) > k)); \\ Antti Karttunen, Sep 30 2025

Crossrefs

Cf. A007947, A053669, A126706, A360765, A360767, A360768, A389216.

Sequence in context: A389227 A364996 A391026 * A270660 A383488 A173490

Adjacent sequences: A363079 A363080 A363081 * A363083 A363084 A363085

Keyword

nonn

Author

Michael De Vlieger, Jul 29 2023

Status

approved