A308935 - OEIS

%I #30 Aug 13 2024 13:45:50

%S 2,8,12,64,18,216,35,112,360,818,660,348,208,2744,693,4096,493,450,

%T 3420,4832,1071,2112,1242,13824,7800,17576,1998,4368,10133,1560,1178,

%U 1280,3597,3060,8582,46656,5032,1292,29640,12768,1189,14868,3182,13112,36468,6670

%N a(n) is the smallest m > n such that n^2*(n^2 + 1) divides m^2*(m^2 + 1).

%C For any n > 0, a(n) exists as n^2*(n^2+1) divides (n^3)^2*((n^3)^2+1).

%C Tsz Ho Chan proved that a(n) >> n*log(n)^(1/8)/log(log(n))^12.

%H Chai Wah Wu, <a href="/A308935/b308935.txt">Table of n, a(n) for n = 1..2376</a>

%H Tsz Ho Chan, <a href="https://arxiv.org/abs/1906.11128">Gaps between divisible terms in a^2*(a^2+1)</a>, arXiv:1906.11128 [math.NT], 2019.

%H Tsz Ho Chan, <a href="https://arxiv.org/abs/2408.01306">The Diophantine equation $b (b+1) (b+2) = t a (a + 1) (a + 2)$ and gap principle</a>, arXiv preprint (2024). arXiv:2408.01306 [math.NT]

%F a(n) <= n^3.

%e For n = 2:

%e - A071253(3) mod A071253(2) = 10,

%e - A071253(4) mod A071253(2) = 12,

%e - A071253(5) mod A071253(2) = 10,

%e - A071253(6) mod A071253(2) = 12,

%e - A071253(7) mod A071253(2) = 10,

%e - A071253(8) mod A071253(2) = 0,

%e - hence a(2) = 8.

%t a[n_] := With[{n2 = n^2(n^2+1)}, For[m = n+1, True, m++, If[Divisible[ m^2(m^2+1), n2], Print[n, " ", m]; Return[m]]]];

%t a /@ Range[100] (* _Jean-François Alcover_, Dec 20 2019 *)

%o (PARI) a(n, f = x->x^2*(x^2+1)) = my (fn=f(n)); for (m=n+1, oo, if (f(m)%fn==0, return (m)))

%o (Python)

%o def A308935(n):

%o     n2, m, m2 = n**2*(n**2+1), n+1, ((n+1)**2*((n+1)**2+1)) % (n**2*(n**2+1))

%o     while m2:

%o         m2, m = (m2 + 2*(2*m+1)*(m**2+m+1)) % n2, (m+1) % n2

%o     return m # _Chai Wah Wu_, Jul 01 2019

%o (Magma) a:=[]; for n in [1..50] do m:=n+1; while not IsIntegral( (m^2*(m^2 + 1))/(n^2*(n^2 + 1) ))  do m:=m+1; end while; Append(~a,m); end for; a; // _Marius A. Burtea_, Dec 20 2019

%Y Cf. A065876, A071253.

%K nonn

%O 1,1

%A _Rémy Sigrist_, Jul 01 2019