A365870 -id:A365870 - cp's OEIS frontend

A365869 Numbers whose exponent of least prime factor in their prime factorization is even.

4, 9, 12, 16, 20, 25, 28, 36, 44, 45, 48, 49, 52, 60, 63, 64, 68, 76, 80, 81, 84, 92, 99, 100, 108, 112, 116, 117, 121, 124, 132, 140, 144, 148, 153, 156, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 204, 207, 208, 212, 220, 225, 228, 236, 240, 244, 252, 256

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that A067029(k) is positive and even.
The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = (1/(prime(n)*(prime(n)+1))) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4, 5 and 6, d(n) = 1/6, 1/24, 1/90, 1/210, 2/1155 and 8/7007.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.229627797346...

			4 is a term since the exponent of the prime factor 2 in the factorization 4 = 2^2 is 2, which is even.

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

Cf. A020639, A067029.
Subsequence of A283050.
Subsequences: A365870, A365871.

Select[Range[256], EvenQ[FactorInteger[#][[1, -1]]] &]

PARI
```
is(n) = n > 1 && !(factor(n)[1,2]%2);
```

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

Original entry on oeis.org

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

Amiram Eldar, Sep 21 2023

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... .

			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.

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

Cf. A067029.

Subsequence of A004767, A365869 and A365870.

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

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, ", ")));

cp's OEIS Frontend

A365869 Numbers whose exponent of least prime factor in their prime factorization is even.

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