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

1,1

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... .

Links

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

Example

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.

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]

Prog

(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); }

Crossrefs

Cf. A067029.

Subsequence of A068140 and A365866.

A365868 is a subsequence.

Sequence in context: A223087 A261549 A068140 * A349307 A349308 A224547

Adjacent sequences: A365864 A365865 A365866 * A365868 A365869 A365870

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved