A365871 Starts of runs of 3 consecutive integers whose exponent of least prime factor in their prime factorization is even.

475, 1519, 2223, 2275, 3283, 4475, 4923, 4975, 5823, 6723, 6811, 7299, 7675, 8107, 8379, 8523, 8955, 9475, 10323, 10467, 11275, 12427, 12463, 12591, 13075, 13867, 13923, 14355, 15631, 15723, 16675, 18027, 18275, 18475, 18767, 19323, 19375, 19647, 22075, 22831

Offset

1,1

Comments

Numbers k such that k, k+1 and k+2 are all terms of A365869.

Numbers of the form 4*k+2 are not terms of A365869. 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 1, 18, 195, 1952, 19542, 195514, 1955859, 19560453, 195611458, ... . Apparently, the asymptotic density of this sequence exists and equals 0.001956... .

Links

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

Example

475 is a term since the exponent of the prime factor 5 in the factorization 475 = 5^2 * 19 is 2, which is even, the exponent of the prime factor 2 in the factorization 476 = 2^2 * 7 * 17 is 2, which is even, and the exponent of the prime factor 3 in the factorization 477 = 3^2 * 53 is also 2, which is even.

Mathematica

Select[4 * Range[6000] + 3, AllTrue[# + {0, 1, 2}, EvenQ[FactorInteger[#1][[1, -1]]] &] &]

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, A365869 and A365870.

Sequence in context: A104649 A165463 A252518 * A210048 A263705 A048428

Adjacent sequences: A365868 A365869 A365870 * A365872 A365873 A365874

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved