A365868 Starts of runs of 3 consecutive integers that are divisible by the cube of their least prime factor.

1375, 16119, 25623, 28375, 52623, 55375, 57967, 79623, 82375, 90207, 94471, 106623, 109375, 129623, 133623, 134431, 136375, 160623, 163375, 164295, 187623, 190375, 206143, 214623, 217375, 241623, 244375, 268623, 271375, 280231, 295623, 298375, 312471, 322623, 325375

Offset

1,1

Comments

Numbers k such that k, k+1 and k+2 are all terms of A365866.

Numbers of the form 4*k+2 are not terms of A365866. Therefore, there are no runs of 4 or more consecutive integers. Since the middle integer in each triple is divisible by 8, all the terms of this sequence are of the form 8*k+7.

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are , 1, 11, 109, 1092, 10899, 109125, 1091335, 10912897, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0001091... .

Links

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

Example

1375 is a term since 5 is the least prime factor of 1375 and 1375 is divisible by 5^3 = 125, 2 is the least prime factor of 1376 and 1376 is divisible by 2^3 = 8, and 3 is the least prime factor of 1377 and 1377 is divisible by 3^3 = 27.

Mathematica

Select[8 * Range[41000] + 7, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 3 &] &]

Prog

(PARI) is(n) = factor(n)[1, 2] >= 3;
lista(kmax) = forstep(k = 7, kmax, 8, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

Crossrefs

Cf. A067029.

Subsequence of A004771, A068140, A365866 and A365867.

Sequence in context: A258964 A349235 A045131 * A056049 A206340 A056087

Adjacent sequences: A365865 A365866 A365867 * A365869 A365870 A365871

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved