cp's OEIS Frontend

A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.

%I A232175 #43 Nov 13 2024 15:02:55
%S A232175 0,1,3,6,10,3,21,8,36,15,55,6,78,35,15,48,136,27,171,10,42,99,253,10,
%T A232175 300,143,81,42,406,15,465,64,88,255,35,63,666,323,91,3,820,21,903,55,
%U A232175 66,483,1081,48,1176,125,85,39,1378,81,165,28,76,783,1711,15,1830,899,63
%N A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.
%C A232175 Numbers n such that a(n) = n*(n-1)/2  appear to be A000430.
%C A232175 n = 1 is the only number for which a(n) = 0. - _T. D. Noe_, Nov 21 2013
%H A232175 Chai Wah Wu, <a href="/A232175/b232175.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1000 from T. D. Noe).
%H A232175 StackExchange, <a href="http://math.stackexchange.com/questions/112561">The cube of integer can be written as the difference of two square</a>.
%t A232175 Join[{0}, Table[k = 1; While[! IntegerQ[Sqrt[n^3 + k^2]], k++]; k, {n, 2, 100}]] (* _T. D. Noe_, Nov 21 2013 *)
%o A232175 (Python)
%o A232175 import math
%o A232175 for n in range(77):
%o A232175    n3 = n*n*n
%o A232175    y=1
%o A232175    for k in range(1, 10000001):
%o A232175      s = n3 + k*k
%o A232175      r = int(math.sqrt(s))
%o A232175      if r*r == s:
%o A232175        print(k, end=', ')
%o A232175        y=0
%o A232175        break
%o A232175    if y: print(end='-, ')
%o A232175 (Python)
%o A232175 from __future__ import division
%o A232175 from sympy import divisors
%o A232175 def A232175(n):
%o A232175     n3 = n**3
%o A232175     ds = divisors(n3)
%o A232175     for i in range(len(ds)//2-1,-1,-1):
%o A232175         x = ds[i]
%o A232175         y = n3//x
%o A232175         a, b = divmod(y-x,2)
%o A232175         if not b:
%o A232175             return a
%o A232175     return 0 # _Chai Wah Wu_, Sep 12 2017
%o A232175 (PARI) a(n) = {k = 1; while (!issquare(n^3+k^2), k++); k;} \\ _Michel Marcus_, Nov 20 2013
%Y A232175 Cf. A000290, A000430, A000578, A038202, A055527, A232176.
%K A232175 nonn,look
%O A232175 1,3
%A A232175 _Alex Ratushnyak_, Nov 19 2013