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

1,1

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... .

Links

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

Index entries for sequences computed from exponents in factorization of n.

Example

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

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]

Prog

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

Crossrefs

Cf. A057521, A060687, A085986, A179119.

Subsequence: A179643.

Sequence in context: A043438 A044223 A044604 * A340017 A303606 A303661

Adjacent sequences: A375141 A375142 A375143 * A375145 A375146 A375147

Keyword

nonn,easy

Author

Amiram Eldar, Aug 01 2024

Status

approved