A365864 Numbers k such that k and k+1 are both divisible by the square of their least prime factor.

8, 24, 27, 44, 48, 63, 80, 99, 116, 120, 124, 135, 152, 168, 171, 175, 188, 207, 224, 243, 260, 275, 279, 288, 296, 315, 324, 332, 343, 351, 360, 368, 387, 404, 423, 424, 440, 459, 475, 476, 495, 512, 528, 531, 539, 548, 567, 575, 584, 603, 620, 624, 636, 639

Offset: 1

Author

Amiram Eldar, Sep 21 2023

Keywords

• nonn
• easy

Comments

Numbers k such that k and k+1 are both terms of A283050.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 8, 82, 802, 8009, 80078, 800900, 8009533, 80097354, 800979764, 8009809838, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08009... .

Examples

			8 is a term since 2 is the least prime factor of 8 and 8 is divisible by 2^2 = 4, and 3 is the least prime factor of 9 and 9 is divisible by 3^3 = 9.
		

Links

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

Crossrefs

Cf. A067029.

Subsequence of A068781 and A283050.

A365865 is a subsequence.

Programs

Mathematica

q[n_] := FactorInteger[n][[1, -1]] >= 2; 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[640]

PARI

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