A365865 Starts of runs of 3 consecutive integers that are divisible by the square of their least prime factor.

423, 475, 1323, 1375, 1519, 2007, 2223, 2275, 2871, 3123, 3175, 3211, 3283, 3479, 3575, 3751, 3771, 4023, 4075, 4475, 4923, 4959, 4975, 5047, 5535, 5823, 5875, 6723, 6775, 6811, 7299, 7623, 7675, 8107, 8379, 8523, 8575, 8955, 9423, 9475, 10323, 10339, 10375, 10467

Offset

1,1

Comments

Numbers k such that k, k+1 and k+2 are all terms of A283050.

Numbers of the form 4*k+2 are not terms of A283050. Therefore, there are no runs of 4 or more consecutive integers, and all the terms of this sequence are of the form 4*k+3.

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 40, 429, 4419, 44352, 444053, 4441769, 44421000, 444220814, ... . Apparently, the asymptotic density of this sequence exists and equals 0.004442... .

Links

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

Example

423 is a term since 3 is the least prime factor of 423 and 423 is divisible by 3^2 = 9, 2 is the least prime factor of 424 and 424 is divisible by 2^2 = 4, and 5 is the least prime factor of 425 and 425 is divisible by 5^2 = 25.

Mathematica

Select[4 * Range[2700] + 3, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 2 &] &]

Prog

(PARI) is(n) = factor(n)[1, 2] >= 2;
lista(kmax) = forstep(k = 3, kmax, 4, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

Crossrefs

Cf. A067029.

Subsequence of A004767, A070258, A283050 and A365864.

Sequence in context: A238285 A231939 A203099 * A348099 A096024 A205980

Adjacent sequences: A365862 A365863 A365864 * A365866 A365867 A365868

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved