

A251728


Semiprimes p*q for which p <= q < p^2.


14



4, 6, 9, 15, 21, 25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 133, 143, 161, 169, 187, 203, 209, 217, 221, 247, 253, 259, 287, 289, 299, 301, 319, 323, 329, 341, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703
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OFFSET

1,1


COMMENTS

Semiprimes p*q for which there exists r <= q such that r^k <= p <= q < r^(k+1), for some k >= 1, i.e., semiprimes whose both prime factors fit inside a semiopen range of two consecutive powers of some natural number r which itself is not greater than the larger prime factor. If such r exists, then it must be <= p (the smaller prime factor of n), which forces q to be less than p^2. On the other hand, when p <= q < p^2, then setting r = p and k = 1 satisfies the equation r^k <= p <= q < r^(k+1).
Assuming that A054272(n), the number of primes in interval [p(n), p(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true), it follows that for any a(n), A003961(a(n)) is also in sequence. In other words, whenever prime(i)*prime(j) is in the sequence, then so is also prime(i+1)*prime(j+1).
From above would follow also that these are all the "settled semiprimes" that occur in a square array A083221 constructed from the sieve of Eratosthenes, from the level A251719 downward. Furthermore, this sequence would then be an infinite disjoint union of sequences of A003961iterates starting from the initial values given in A251724.
See also the comments in the complementary sequence of semiprimes, A138511.
Composite numbers n with all prime factors greater than the cube root of n.  Doug Bell, Oct 27 2015
If "p <= q" in the definition were changed to "p < q" then the squares of primes (A001248) would be removed, yielding A138109.  Jon E. Schoenfield, Dec 27 2022


LINKS



FORMULA



MATHEMATICA

fQ[n_] := Block[{pf = FactorInteger@ n, p, q}, p = pf[[1, 1]]; q = pf[[1, 1]]; And[p <= q < p^2, PrimeOmega@ n == 2]]; Select[Range@ 720, fQ] (* Michael De Vlieger, Oct 27 2015 *)


PROG

(Scheme, with Antti Karttunen's IntSeqlibrary, two alternative definitions)
(Haskell)
a251728 n = a251728_list !! (n1)
a251728_list = filter f [1..] where
f x = q < p ^ 2 && a010051' q == 1
where q = div x p; p = a020639 x
(PARI) lista(nn) = forcomposite(n=1, nn, my(f = factor(n)); if (#select(x>(x > n^(1/3)), f[, 1]) == #f~, print1(n, ", "))); \\ Michel Marcus, Oct 27 2015
(PARI) list(lim)=my(v=List()); forprime(q=2, sqrtnint((lim\1)^2, 3), forprime(p=sqrtint(q)+1, min(q, lim\q), listput(v, p*q))); Set(v) \\ Charles R Greathouse IV, Oct 27 2015


CROSSREFS

A251724 after the initial 2 is a subsequence.


KEYWORD

nonn


AUTHOR



STATUS

approved



