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%I #17 Jan 03 2024 09:38:43
%S 0,2,3,6,12,20,24,30,40,42,56,60,68,75,78,84,87,120,126,160,180,248,
%T 264,270,273,308,312,318,330,336,351,360,396,564,570,588,615,620,630,
%U 635,720,738,780,840,912,1008,1016,1032,1284,1308,1320,1334,1344,1404,1540,1617
%N Numbers n such that m - n^3 is a square, where m is the least square above n^3.
%C Numbers n such that A070929(n) is a nonzero square.
%C The sequence of cubes a(n)^3 begins: 0, 8, 27, 216, 1728, 8000, 13824, 27000, 64000, 74088, 175616, 216000, 314432, ...
%C The sequence of m's begins: 1, 9, 36, 225, 1764, 8100, 13924, 27225, 64009, 74529, 176400, 216225, 314721, ...
%C The sequence of square roots of these m's begins: 1, 3, 6, 15, 42, 90, 118, 165, 253, 273, 420, 465, 561, 650, 689, 770, 812, ...
%C The sequence of squares m-n^3 begins: 1, 1, 9, 9, 36, 100, 100, 225, 9, 441, 784, 225, 289, 625, 169, 196, 841, ...
%C The sequence of their square roots begins: 1, 1, 3, 3, 6, 10, 10, 15, 3, 21, 28, 15, 17, 25, 13, 14, 29, 35, 43, 24, ... (note the first 12 terms are triangular numbers, A000217).
%H Charles R Greathouse IV, <a href="/A233422/b233422.txt">Table of n, a(n) for n = 1..10000</a>
%t fQ[n_]:=Module[{c=n^3,m},m=(Floor[Sqrt[c]]+1)^2;IntegerQ[Sqrt[m-c]]];Select[Range[0,1650],fQ] (* _Harvey P. Dale_, Jan 03 2024 *)
%o (Python)
%o def isqrt(a):
%o sr = 1L << (long.bit_length(long(a)) >> 1)
%o while a < sr*sr: sr>>=1
%o b = sr>>1
%o while b:
%o s = sr+b
%o if a >= s*s: sr = s
%o b>>=1
%o return sr
%o def isSquare(a):
%o sr = isqrt(a)
%o return (a==sr*sr)
%o for n in range(77777):
%o n3 = n*n*n
%o a = isqrt(n3)+1
%o if isSquare(a*a-n3): print str(n)+',',
%o (PARI) is(n)=issquare((sqrtint(n=n^3)+1)^2-n) \\ _Charles R Greathouse IV_, Dec 09 2013
%Y Cf. A000290, A000578, A070929, A077115, A154101, A233400, A233401.
%K nonn
%O 1,2
%A _Alex Ratushnyak_, Dec 09 2013
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