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

Author

Amiram Eldar, Sep 21 2023

Keywords

• nonn
• easy

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... .

Examples

			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.
		

Links

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

Crossrefs

Cf. A067029.

Subsequence of A004767, A070258, A283050 and A365864.

Programs

Mathematica

Select[4 * Range[2700] + 3, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 2 &] &]
SequencePosition[Table[If[Divisible[n,FactorInteger[n][[1,1]]^2],1,0],{n,11000}],{1,1,1}][[;;,1]] (* Harvey P. Dale, Aug 05 2024 *)

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, ", ")));