A364999 Numbers k neither squarefree nor prime power such that both rad(k)*A119288(k) > k and rad(k)*A053669(k) > k.

12, 20, 28, 44, 52, 60, 68, 76, 84, 92, 116, 124, 132, 140, 148, 156, 164, 172, 188, 204, 212, 220, 228, 236, 244, 260, 268, 276, 284, 292, 308, 316, 332, 340, 348, 356, 364, 372, 380, 388, 404, 412, 420, 428, 436, 444, 452, 460, 476, 492, 508, 516, 524, 532, 548

Offset

1,1

Comments

Subset of A126706, numbers that are neither squarefree nor prime powers.

For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).

A355432(k) = A360543(k) = 0. There exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k.

Apparently this is A081770 without the leading 4. - R. J. Mathar, Sep 05 2023

From Peter Munn, Mar 05 2024: (Start)

The preceding observation is true for the whole sequence, for reasons outlined below.

To qualify for this sequence, a number k must be smaller than 2 different multiples of rad(k): one based on a divisor, A119288(k): the other on a nondivisor, A053669(k).

For k that is not a prime power, straightforward calculations show (1) if k = 2 * rad(k) then k satisfies both of these comparisons, whereas (2) for k >= 3 * rad(k), k fails the divisor-based comparison if k is a multiple of 6 and fails the nondivisor-based comparison otherwise.

(End)

Links

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

Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364998 in gold, in A364997 in green, and in A361098 in red.

Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms. This uses the same color scheme as described immediately above.

Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates fairly constant density of a(n) in A126706 as well as a slight quasiperiodic pattern approximately mod 169.

Formula

Intersection of A363082 and A360767.

From Peter Munn, Feb 21 2024: (Start)

a(n) = 2*A039956(n+1).

Asymptotic density is 1/Pi^2 = 0.101321183642337... (A092742). (End)

From Michael De Vlieger, Mar 08 2024: (Start)

{a(n)} = A366825 \ A366460, i.e., even terms in A366825.

A088860 = {a(n)} intersect A025487 = {a(n)} intersect A055932, where A088860(k) = 2*A002110(k). (End)

Example

Let b(n) = A126706(n), S = A360767, and T = A363082.
b(1) = a(1) = 12 since p*r = 3*6 = 18 and q*r = 5*6 = 30, and both exceed 12. Indeed, 12 is in both S and T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18; 18 is not in S.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, and is not in T, etc.

Mathematica

Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Crossrefs

Cf. A007947, A039956, A053669, A081770, A088860, A092742, A119288, A126706, A355432, A360432, A360767, A361098, A363082, A364998, A364999.

Sequence in context: A366825 A110187 A096156 * A366807 A210968 A107277

Adjacent sequences: A364996 A364997 A364998 * A365000 A365001 A365002

Keyword

nonn

Author

Michael De Vlieger, Aug 16 2023

Status

approved