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

Author

Amiram Eldar, Sep 21 2023

Keywords

• nonn
• easy

Comments

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...

Examples

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

Links

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

Crossrefs

Cf. A020639, A067029.

Subsequence of A283050.

Subsequences: A365870, A365871.

Programs

Mathematica

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

PARI

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