%I #7 Nov 03 2014 01:17:30

%S 2,4,14,34,254,664,5116,18446,121694,887314,7496644,63124214,

%T 684394346,3086525014,25689944554,453164666954

%N Smallest k such that k^2-1 is a squarefree number with n prime divisors. a(n) = A088027(n)^(1/2).

%e a(4)^2 = 1156 = 34^2 = 3*5*7*11 + 1.

%o (Scheme program from Thomas Baruchel); (define primes '(2 3 5 7 ... 999983)) (compute n) returns A088028(n) (or #f if prime list is too short) computation takes a reasonable amount of time for n <= 16 (slower than "brutal" method for small values of n, but soon becomes much quicker). Result is certified to be the smallest.

%o (define (compute* n mmax prod offset) (do ((i offset (+ i 1)) (l (length primes))) ((>= (* prod (do ((j 0 (+ j 1)) (p 1)) ((= j n) p) (set! p (* p (list-ref primes (+ i j)))))) mmax) mmax) (let ((p (* prod (list-ref primes i)))) (if (> n 1) (set! mmax (compute* (- n 1) mmax p (+ i 1))) (let ((s (inexact->exact (floor (sqrt (+ p 1)))))) (if (= (* s s) (+ p 1)) (set! mmax p)))))))

%o (define (compute n) (let* ((p (reverse (cdr primes))) (mmax (apply * (cons (car p) (list-tail p (- (length p) (- n 1)))))) (r (compute* n mmax 1 1))) (if (= mmax r) #f (inexact->exact (floor (sqrt (+ r 1)))))))

%Y Cf. A088027.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Sep 19 2003

%E More terms from _Ray Chandler_, Oct 04 2003

%E Further terms from _Thomas Baruchel_, Oct 11 2003