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

1,1

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... .

Links

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

Example

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.

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]

Prog

(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); }

Crossrefs

Cf. A067029.

Subsequence of A068781 and A283050.

A365865 is a subsequence.

Sequence in context: A175496 A048109 A068781 * A212861 A333961 A038524

Adjacent sequences: A365861 A365862 A365863 * A365865 A365866 A365867

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved