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

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, 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, 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

Offset

1,4

Comments

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.

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10351

Formula

Equally: a(1) = a(2) = a(3) = 1; and for n>=4: a(n) = the largest k such that A250474(k-1) <= n.

Other identities. For all n >= 1:

a(n) >= A251718(n) >= A251717(n).

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.]

Prog

(Scheme, two variants)
(define (A251719 n) (let loop ((k 1)) (if (> (A250474 k) n) k (loop (+ 1 k))))) ;; Very straightforward version.
;; Code for A083221bi given in A083221.
(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))))))

Crossrefs

Cf. A001222, A003961, A055396, A083221, A250474, A243056, A251717, A251718, A251723, A251724.

Sequence in context: A090529 A297212 A155934 * A130822 A194220 A189627

Adjacent sequences: A251716 A251717 A251718 * A251720 A251721 A251722

Keyword

nonn

Author

Antti Karttunen, Dec 15 2014

Status

approved