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A251724 a(1) = 2, and for n>1: a(n) = prime(A251719(n)) * prime(A251719(n) + n - 2), where prime(n) gives the n-th prime.

%I A251724 #27 Mar 05 2022 14:17:42
%S A251724 2,4,6,21,65,85,95,115,217,259,287,301,329,649,671,737,781,803,869,
%T A251724 913,979,1067,1111,1133,1177,1199,1243,1703,1781,1807,1937,1963,2041,
%U A251724 2119,2171,3043,3077,3247,3281,3349,3383,3587,3791,3859,3893,3961,4063,4097,4267,4369,4471,4573,4607,4709,4777,4811,5833,5909,5947,6023,6289,6403,6593,6631,6707,6821,8579
%N A251724 a(1) = 2, and for n>1: a(n) = prime(A251719(n)) * prime(A251719(n) + n - 2), where prime(n) gives the n-th prime.
%C A251724 For n >= 2: a(n) = the first "settled semiprime" in the column n of the sieve of Eratosthenes: a(n) = A083221(A251719(n), n).
%C A251724 The "settling of semiprimes" here means that from that semiprime onward, all the other terms in the same column n of a square array A083221 (which is constructed from the sieve of Eratosthenes) are also semiprimes, obtained by successive iterations of A003961 starting from the semiprime here given as a(n). Cf. comments in A251728 which contains all such semiprimes. The "unsettled" semiprimes are in its complement A138511.
%C A251724 Here we assume that A054272(n), the number of primes in interval [prime(n), prime(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true).
%H A251724 Antti Karttunen, <a href="/A251724/b251724.txt">Table of n, a(n) for n = 1..10351</a>
%H A251724 Wikipedia, <a href="http://en.wikipedia.org/wiki/Brocard%27s_conjecture">Brocard's Conjecture</a>
%F A251724 a(1) = 2; and for n >= 2: a(n) = A000040(A251719(n)) * A000040(A251719(n) + n - 2).
%F A251724 a(n) = A083221(A251719(n), n).
%F A251724 Other identities implied by the definition. For all n >= 1:
%F A251724 A078898(a(n)) = n.
%F A251724 A055396(a(n)) = A251719(n).
%F A251724 For all n >= 2:
%F A251724 A243055(a(n)) = n-2.
%o A251724 (Scheme, two versions)
%o A251724 (define (A251724 n) (if (= 1 n) 2 (* (A000040 (A251719 n)) (A000040 (+ (A251719 n) n -2)))))
%o A251724 (define (A251724 n) (A083221bi (A251719 n) n))
%Y A251724 After initial 2, a subsequence of A251728 and A001358.
%Y A251724 Cf. A000040, A003961, A054272, A055396, A078898, A083221, A138511, A243055, A251719.
%K A251724 nonn
%O A251724 1,1
%A A251724 _Antti Karttunen_, Dec 15 2014