A365868 -id:A365868 - cp's OEIS frontend

A365866 Integers that are divisible by the cube of their least prime factor.

8, 16, 24, 27, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 144, 152, 160, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 360, 368, 376

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that A067029(k) >= 3.
The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = (1/prime(n)^3) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4, 5 and 6, d(n) = 1/8, 1/54, 1/375, 4/5145, 8/46585 and 16/169169.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.147333958520714364623977...

			8 is a term since 2 is the least prime factor of 8 and 8 is divisible by 2^3 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020639, A067029.
Subsequence of A046099 and A283050.
Subsequences: A365867, A365868.

Select[Range[400], FactorInteger[#][[1, -1]] >= 3 &]

PARI
```
is(n) = n > 1 && factor(n)[1,2] >= 3;
```

A365867 Numbers k such that k and k+1 are both divisible by the cube of their least prime factor.

Original entry on oeis.org

80, 135, 296, 343, 351, 512, 567, 624, 728, 783, 944, 999, 1160, 1215, 1375, 1376, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2079, 2240, 2295, 2375, 2400, 2456, 2511, 2672, 2727, 2888, 2943, 3104, 3159, 3320, 3375, 3536, 3591, 3624, 3752, 3807, 3968, 4023, 4184

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that k and k+1 are both terms of A365866.

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are , 12, 110, 1119, 11167, 111662, 1116693, 11166978, 111669826, 1116697990, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0111669... .

			80 is a term since 2 is the least prime factor of 80 and 80 is divisible by 2^3 = 8, and 3 is the least prime factor of 81 and 81 is divisible by 3^3 = 27.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A067029.
Subsequence of A068140 and A365866.
A365868 is a subsequence.

q[n_] := FactorInteger[n][[1, -1]] >= 3; consec[kmax_] := Module[{m = 1, c = Table[False, {2}], s = {}}, Do[c = Join[Rest[c], {q[k]}]; If[And @@ c, AppendTo[s, k - 1]], {k, 1, kmax}]; s]; consec[5000]

lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = factor(k)[1,2] >= 3; if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

cp's OEIS Frontend

A365866 Integers that are divisible by the cube of their least prime factor.

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