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Semiprimes p*q for which p <= q < p^2.
14

%I #55 Dec 28 2022 02:14:32

%S 4,6,9,15,21,25,35,49,55,65,77,85,91,95,115,119,121,133,143,161,169,

%T 187,203,209,217,221,247,253,259,287,289,299,301,319,323,329,341,361,

%U 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

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

%C 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).

%C 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).

%C 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 A003961-iterates starting from the initial values given in A251724.

%C See also the comments in the complementary sequence of semiprimes, A138511.

%C Composite numbers n with all prime factors greater than the cube root of n. - _Doug Bell_, Oct 27 2015

%C 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

%H Antti Karttunen, <a href="/A251728/b251728.txt">Table of n, a(n) for n = 1..10000</a>

%H Carlos Rivera, <a href="https://www.primepuzzles.net/conjectures/conj_090.htm">Conjecture 90. Semiprimes/Primes ->log(2)</a>, The Prime Puzzles & Problems Connection.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Brocard%27s_conjecture">Brocard's conjecture</a>

%F For all n >= 1, A078898(a(n)) = A243055(a(n)) + 2.

%F Limit_{n->oo} n*log(a(n))/a(n) = log(2). - _Alain Rocchelli_, Nov 10 2022

%t 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 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library, two alternative definitions)

%o (define A251728 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (< (A006530 n) (A000290 (A020639 n)))))))

%o (define A251728 (COMPOSE A001358 (MATCHING-POS 1 1 (lambda (n) (<= (A138510 n) (A006530 (A001358 n)))))))

%o (Haskell)

%o a251728 n = a251728_list !! (n-1)

%o a251728_list = filter f [1..] where

%o f x = q < p ^ 2 && a010051' q == 1

%o where q = div x p; p = a020639 x

%o -- _Reinhard Zumkeller_, Jan 06 2015

%o (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

%o (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

%Y An intersection of A251726 and A001358 (semiprimes).

%Y Complement of A138511 in A001358.

%Y A251724 after the initial 2 is a subsequence.

%Y Cf. A003961, A006530, A054272, A078898, A083221, A138109, A138510, A243055, A250474, A251719.

%K nonn

%O 1,1

%A _Antti Karttunen_, Dec 16 2014