cp's OEIS Frontend

A251728 Semiprimes p*q for which p <= q < p^2.

%I A251728 #60 Mar 05 2025 10:39:52
%S A251728 4,6,9,15,21,25,35,49,55,65,77,85,91,95,115,119,121,133,143,161,169,
%T A251728 187,203,209,217,221,247,253,259,287,289,299,301,319,323,329,341,361,
%U A251728 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 A251728 Semiprimes p*q for which p <= q < p^2.
%C A251728 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 A251728 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 A251728 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 A251728 See also the comments in the complementary sequence of semiprimes, A138511.
%C A251728 Composite numbers n with all prime factors greater than the cube root of n. - _Doug Bell_, Oct 27 2015
%C A251728 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 A251728 Antti Karttunen, <a href="/A251728/b251728.txt">Table of n, a(n) for n = 1..10000</a>
%H A251728 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 A251728 Wikipedia, <a href="http://en.wikipedia.org/wiki/Brocard%27s_conjecture">Brocard's conjecture</a>
%F A251728 For all n >= 1, A078898(a(n)) = A243055(a(n)) + 2.
%F A251728 Limit_{n->oo} n*log(a(n))/a(n) = log(2). - _Alain Rocchelli_, Nov 10 2022
%t A251728 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 A251728 (Scheme, with _Antti Karttunen_'s IntSeq-library, two alternative definitions)
%o A251728 (define A251728 (MATCHING-POS 1 2 (lambda (n) (and (= 2 (A001222 n)) (< (A006530 n) (A000290 (A020639 n)))))))
%o A251728 (define A251728 (COMPOSE A001358 (MATCHING-POS 1 1 (lambda (n) (<= (A138510 n) (A006530 (A001358 n)))))))
%o A251728 (Haskell)
%o A251728 a251728 n = a251728_list !! (n-1)
%o A251728 a251728_list = filter f [1..] where
%o A251728                       f x = q < p ^ 2 && a010051' q == 1
%o A251728                             where q = div x p; p = a020639 x
%o A251728 -- _Reinhard Zumkeller_, Jan 06 2015
%o A251728 (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 A251728 (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
%o A251728 (Python)
%o A251728 from math import isqrt
%o A251728 from sympy import primepi, primerange
%o A251728 def A251728(n):
%o A251728     def bisection(f,kmin=0,kmax=1):
%o A251728         while f(kmax) > kmax: kmax <<= 1
%o A251728         kmin = kmax >> 1
%o A251728         while kmax-kmin > 1:
%o A251728             kmid = kmax+kmin>>1
%o A251728             if f(kmid) <= kmid:
%o A251728                 kmax = kmid
%o A251728             else:
%o A251728                 kmin = kmid
%o A251728         return kmax
%o A251728     def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(min(x//p,p**2)) for p in primerange(s+1)))
%o A251728     return bisection(f,n,n) # _Chai Wah Wu_, Mar 05 2025
%Y A251728 An intersection of A251726 and A001358 (semiprimes).
%Y A251728 Complement of A138511 in A001358.
%Y A251728 A251724 after the initial 2 is a subsequence.
%Y A251728 Cf. A003961, A006530, A054272, A078898, A083221, A138109, A138510, A243055, A250474, A251719.
%K A251728 nonn
%O A251728 1,1
%A A251728 _Antti Karttunen_, Dec 16 2014