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

1,1

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...

Links

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A020639, A067029.

Subsequence of A283050.

Subsequences: A365870, A365871.

Sequence in context: A337103 A312855 A348752 * A312856 A010449 A210607

Adjacent sequences: A365866 A365867 A365868 * A365870 A365871 A365872

Keyword

nonn,easy

Author

Amiram Eldar, Sep 21 2023

Status

approved