A375144 Numbers whose prime factorization has exactly two exponents that equal 2 and has no higher exponents.

36, 100, 180, 196, 225, 252, 300, 396, 441, 450, 468, 484, 588, 612, 676, 684, 700, 828, 882, 980, 1044, 1089, 1100, 1116, 1156, 1225, 1260, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1692, 1700, 1900, 1908, 1980, 2028, 2100, 2116, 2124, 2156, 2178, 2196

Offset: 1

Author

Amiram Eldar, Aug 01 2024

Keywords

• nonn
• easy

Comments

Numbers of the form m * p^2 * q^2, where p < q are primes, and m is a squarefree number such that gcd(m, p*q) = 1.

Numbers whose powerful part (A057521) is a square of a squarefree semiprime (A085986).

The asymptotic density of this sequence is ((Sum_{p prime} 1/(p*(p+1)))^2 - Sum_{p prime} 1/(p*(p+1))^2)/(2*zeta(2)) = 0.022124574473271163980012... .

Examples

			36 = 2^2 * 3^2 is a term since its prime factorization has exactly two exponents and both are equal to 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A057521, A060687, A085986, A179119.

Subsequence: A179643.

Programs

Mathematica

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]], Greater]}, Length[e] > 1 && e[[1;;2]] == {2, 2} && If[Length[e] > 2, e[[3]] == 1, True]]; Select[Range[2200], q]

PARI

is(k) = {my(e = vecsort(factor(k)[,2], , 4)~); #e > 1 && e[1..2] == [2,2] && if(#e > 2, e[3] == 1, 1);}