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Positive integers k whose square is, for some prime p, the largest integer m such that both m and m-1 factor into primes less than or equal to p.
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%I #17 Mar 20 2015 21:03:55

%S 3,9,99,351,13311,1205645,138982582999

%N Positive integers k whose square is, for some prime p, the largest integer m such that both m and m-1 factor into primes less than or equal to p.

%C a(n)^2-1 and a(n)^2 form the largest pair of consecutive p-smooth numbers.

%C Terms are the square roots of square values of A117581(=A002072+1).

%C The corresponding primes p are in A250298.

%e Here are the largest pairs of consecutive integers with prime factors p or smaller:

%e p : pair

%e --------------------------

%e 3 : 3^2-1 and 3^2;

%e 5 : 9^2-1 and 9^2;

%e 11 : 99^2-1 and 99^2;

%e 13 : 351^2-1 and 351^2;

%e 29 : 13311^2-1 and 13311^2;

%e 53 : 1205645^2-1 and 1205645^2;

%e 103 : 138982582999^2-1 and 138982582999^2.

%o (PARI) lista(v_002072) = {v = v_002072; for (i=1, #v, vi = v[i]; if (issquare(vi+1), print1(sqrtint(vi+1), ", ")););} \\ _Michel Marcus_, Feb 28 2015

%Y Cf. A002072, A117581, A250298.

%K hard,more,nonn

%O 1,1

%A _Don N. Page_, Jan 15 2015