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A251719 a(n) = the least k such that A250474(k) > n.

%I A251719 #23 Jan 04 2015 22:54:20
%S A251719 1,1,1,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,
%T A251719 6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,
%U A251719 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10
%N A251719 a(n) = the least k such that A250474(k) > n.
%C A251719 Provided that A250474 is strictly increasing (implied for example if either Legendre's or Brocard's conjecture is true) then all natural numbers occur in this sequence, in order, and after three 1's, each n+1 appears for the first time at A250474(n). Thus from n=2 onward, each n occurs A251723(n-1) times.
%C A251719 With the same provision, we have for n>1: a(n) = smallest positive integer k such that A083221(k, n) is a semiprime and A083221(k+1, n) = A003961(A083221(k, n)), where A003961 shifts the prime factorization one step towards larger primes, thus the latter value is also a semiprime.
%H A251719 Antti Karttunen, <a href="/A251719/b251719.txt">Table of n, a(n) for n = 1..10351</a>
%F A251719 Equally: a(1) = a(2) = a(3) = 1; and for n>=4: a(n) = the largest k such that A250474(k-1) <= n.
%F A251719 Other identities. For all n >= 1:
%F A251719 a(n) >= A251718(n) >= A251717(n).
%F A251719 a(n) = A055396(A251724(n)), or equally, A251724(n) = A083221(a(n), n). [This sequence gives the row-index of the first "settled semiprime" in column n of the sieve of Eratosthenes.]
%o A251719 (Scheme, two variants)
%o A251719 (define (A251719 n) (let loop ((k 1)) (if (> (A250474 k) n) k (loop (+ 1 k))))) ;; Very straightforward version.
%o A251719 ;; Code for A083221bi given in A083221.
%o A251719 (define (A251719 n) (if (= 1 n) 1 (let loop ((i 1) (eka (A083221bi 1 n)) (toka (A083221bi 2 n))) (if (and (= (A001222 eka) 2) (= toka (A003961 eka))) i (loop (+ i 1) toka (A083221bi (+ i 2) n))))))
%Y A251719 Cf. A001222, A003961, A055396, A083221, A250474, A243056, A251717, A251718, A251723, A251724.
%K A251719 nonn
%O A251719 1,4
%A A251719 _Antti Karttunen_, Dec 15 2014