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

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

Author

Amiram Eldar, Sep 21 2023

Keywords

• nonn
• easy

Comments

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... .

Examples

			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.
		

Links

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

Crossrefs

Cf. A067029.

Subsequence of A068140 and A365866.

A365868 is a subsequence.

Programs

Mathematica

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]

PARI

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