A375142 Numbers whose powerful part (A057521) is a power of a squarefree number that is larger than 1 (A072777).

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Author

Amiram Eldar, Aug 01 2024

Keywords

• nonn
• easy

Comments

Subsequence of A013929 and first differs from it at n = 27: A013929(27) = 72 = 2^3 * 3^2 is not a term of this sequence.

Numbers whose prime factorization has one distinct exponent that does not equal 1.

Numbers that are a product of a squarefree number (A005117) and a power of a different squarefree number that is not squarefree.

The asymptotic density of this sequence is Sum_{k>=2} (d(k)-1)/zeta(2) = 0.36113984820338109927..., where d(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

Examples

			12 = 2^2 * 3 is a term because its powerful part, 4 = 2^2, is a power of a squarefree number, 2, that is larger than 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, Vol. 12, No. 1 (1962), pp. 77-84.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A005117, A057521.

Subsequence of A013929.

Subsequences: A067259, A072777, A190641, A336591.

Programs

Mathematica

q[n_] := Count[Union[FactorInteger[n][[;; , 2]]], _?(# > 1 &)] == 1; Select[Range[200], q]

PARI

is(k) = {my(e = select(x -> (x > 1), Set(factor(k)[,2]))); #e == 1;}