A092487 - OEIS

%I #31 Dec 10 2022 10:48:17

%S 0,4,5,0,5,6,7,7,0,8,11,8,13,7,9,0,17,9,19,10,7,11,23,8,0,13,8,12,29,

%T 12,31,13,11,17,13,0,37,19,13,10,41,14,43,11,15,23,47,6,0,13,17,13,53,

%U 16,11,16,19,29,59,15,61,31,14,0,13,14,67,17

%N a(n) = least k such that {n+1, n+2, n+3, ... n+k} has a subset the product of whose members with n is a square.

%C For n>1, n + a(n) is composite and n + a(n) is square if and only if n is square. - _David A. Corneth_, Oct 22 2016

%D R. K. Guy, Unsolved Problems in Number Theory, B30.

%H Peter Kagey, <a href="/A092487/b092487.txt">Table of n, a(n) for n = 1..10000</a>

%H H. B. Bui, K. Pratt, and A. Zaharescu, <a href="https://arxiv.org/abs/2211.12467">A problem of Erdos-Graham-Granville-Selfridge on integral points on hyperelliptic curves</a>, arXiv:2211.12467 [math.NT], 2022.

%H A. Granville and J. L. Selfridge, <a href="https://doi.org/10.37236/1549">Products of integers in an interval, modulo squares</a>, Electron. J. Combin. 8 (2001), no. 1.

%F a(n) = A006255(n) - n. - _Peter Kagey_, Oct 22 2016

%F a(n^2) = 0, a(p) = p for prime p > 3. - _David A. Corneth_, Oct 22 2016

%e a(48)=6 because 48*(48+2)*(48+6) is a square, but you can't square 48 with numbers from (48+1) to (48+5).

%t Table[k = 0; Which[IntegerQ@ Sqrt@ n, k, And[PrimeQ@ n, n > 3], k = n, True, While[Length@ Select[n Map[Times @@ # &, n + Rest@ Subsets@ Range@ k], IntegerQ@ Sqrt@ # &] == 0, k++]]; k, {n, 40}] (* _Michael De Vlieger_, Oct 26 2016 *)

%t Table[k = 0; Which[IntegerQ@Sqrt@n, k, First@Last@FactorInteger@n > Sqrt[2 n] + 1, k = First@Last@FactorInteger@n, True, While[Length@Select[n Map[Times @@ # &, n + Rest@Subsets@Range@k], IntegerQ@Sqrt@# &] == 0, k++]]; k, {n, 100}] (* _Joshua Stucky_, December 5 2022 *)

%o (PARI) a(n) = {if(issquare(n),return(0)); if(isprime(n),if(n>3, return(n), return(n+2) )); my(l = List([n,n+1]), m=2); while(1, for(i=1, #l-2, forvec(v = vector(i, j, [2,#l-1]), if(issquare(l[1] * l[#l] * prod(k=1, #v, l[v[k]])), return(l[#l] - n)), 2)); listput(l,n+m);m++)} \\ _David A. Corneth_, Oct 22 2016

%Y Cf. A006255, A092488.

%K nonn,easy

%O 1,2

%A _Don Reble_, Apr 03 2004