

A251724


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


5



2, 4, 6, 21, 65, 85, 95, 115, 217, 259, 287, 301, 329, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1703, 1781, 1807, 1937, 1963, 2041, 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
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OFFSET

1,1


COMMENTS

For n >= 2: a(n) = the first "settled semiprime" in the column n of the sieve of Eratosthenes: a(n) = A083221(A251719(n), n).
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.
Here we assume 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).


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10351
Wikipedia, Brocard's Conjecture


FORMULA

a(1) = 2; and for n >= 2: a(n) = A000040(A251719(n)) * A000040(A251719(n) + n  2).
a(n) = A083221(A251719(n), n).
Other identities implied by the definition. For all n >= 1:
A078898(a(n)) = n.
A055396(a(n)) = A251719(n).
For all n >= 2:
A243055(a(n)) = n2.


PROG

(Scheme, two versions)
(define (A251724 n) (if (= 1 n) 2 (* (A000040 (A251719 n)) (A000040 (+ (A251719 n) n 2)))))
(define (A251724 n) (A083221bi (A251719 n) n))


CROSSREFS

After initial 2, a subsequence of A251728 and A001358.
Cf. A000040, A003961, A054272, A055396, A078898, A083221, A138511, A243055, A251719.
Sequence in context: A193774 A241210 A176652 * A326363 A273522 A227626
Adjacent sequences: A251721 A251722 A251723 * A251725 A251726 A251727


KEYWORD

nonn


AUTHOR

Antti Karttunen, Dec 15 2014


STATUS

approved



