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A067872 Least m > 0 for which m*n^2 + 1 is a square.

%I A067872 #30 Apr 13 2025 16:23:01
%S A067872 3,2,7,3,23,8,47,15,79,24,119,2,167,48,3,63,287,80,359,6,88,120,527,
%T A067872 28,623,168,727,12,839,44,959,255,216,288,8,20,1367,360,19,77,1679,22,
%U A067872 1847,30,208,528,2207,7,2399,624,128,42,2807,728,696,3,160,840,3479,11
%N A067872 Least m > 0 for which m*n^2 + 1 is a square.
%C A067872 Least m > 0 for which x^2 - m*y^2 = 1 has a solution with y = n.
%C A067872 For n > 1, a(n) <= n^2-2. - _Chai Wah Wu_, Jan 26 2016
%C A067872 Considering the greatest prime factor (gpf) of terms, two curves are evident.  One has gpf ~ n^2 (see a(9949)), the other has gpf ~ n^2/7 (see a(9923)). See the gpf plot link. - _Bill McEachen_, Apr 13 2025
%H A067872 T. D. Noe and Chai Wah Wu, <a href="/A067872/b067872.txt">Table of n, a(n) for n = 1..10000</a> n = 1..500 from T. D. Noe
%H A067872 Bill McEachen, <a href="/A067872/a067872.png">Plot of gpf(sequence terms)</a>
%F A067872 For n a power of an odd prime, a(n) = n^2 - 2. For n twice a power of an odd prime, a(n) = (n/2)^2 - 1. - _T. D. Noe_, Sep 13 2007
%e A067872 a(4)=3, based on 3*4^2 + 1 = 7^2.
%t A067872 a[n_] := For[m=1, True, m++, If[IntegerQ[Sqrt[m*n^2+1]], Return[m]]]; Table[a[n], {n, 100}]
%t A067872 lm[n_]:=Module[{m=1},While[!IntegerQ[Sqrt[m n^2+1]],m++];m]; Array[lm,60] (* _Harvey P. Dale_, Feb 24 2013 *)
%o A067872 (Haskell)
%o A067872 a067872 n = (until ((== 1) . a010052 . (+ 1)) (+ nn) nn) `div` nn
%o A067872             where nn = n ^ 2
%o A067872 -- _Reinhard Zumkeller_, Jun 28 2013
%o A067872 (Python)
%o A067872 def A067872(n):
%o A067872     y, x, n2 = n*(n+2), 2*n+3, n**2
%o A067872     m, r = divmod(y,n2)
%o A067872     while r:
%o A067872         y += x
%o A067872         x += 2
%o A067872         m, r = divmod(y,n2)
%o A067872     return m # _Chai Wah Wu_, Jan 25 2016
%Y A067872 Cf. A033318, A068310.
%Y A067872 Cf. A010052.
%K A067872 nice,nonn
%O A067872 1,1
%A A067872 _Lekraj Beedassy_, Feb 25 2002
%E A067872 Edited by _Dean Hickerson_, Mar 19 2002