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A375144
Numbers whose prime factorization has exactly two exponents that equal 2 and has no higher exponents.
1
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... .
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
Subsequence: A179643.
Sequence in context: A043438 A044223 A044604 * A340017 A303606 A303661
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Aug 01 2024
STATUS
approved