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A274035 Numbers k such that k^7 = a^2 + b^3 for positive integers a and b.

%I A274035 #29 Jun 18 2025 11:52:56
%S A274035 2,5,8,9,10,12,15,17,24,26,28,31,33,36,37,40,43,44,46,50,52,54,56,57,
%T A274035 63,65,68,69,72,73,76,80,82,89,91,98,100,101,108,113,122,126,127,128,
%U A274035 129,134,136,141,145,148,150,152,161,164,168,170,171,174,177,183,185,189,192,196,197
%N A274035 Numbers k such that k^7 = a^2 + b^3 for positive integers a and b.
%H A274035 Jean-François Alcover, <a href="/A274035/b274035.txt">Table of n, a(n) for n = 1..3001</a> (all terms from Charles R Greathouse IV except for a(58)=174)
%H A274035 Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, <a href="http://arxiv.org/abs/math/0508174">Twists of X(7) and primitive solutions to x^2+y^3=z^7</a>, arXiv:math/0508174 [math.NT], 2005; Duke Math. J. 137:1 (2007), pp. 103-158.
%t A274035 okQ[n_] := Module[{a, b}, For[b = 1, b < n^(7/3), b++, If[IntegerQ[a = Sqrt[n^7 - b^3]] && a > 0, Print["n = ", n, ", a = ", a, ", b = ", b]; Return[True]]]; False];
%t A274035 Reap[For[n = 1, n < 200, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Jan 30 2019 *)
%o A274035 (PARI) isA055394(n)=for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0
%o A274035 is(n)=isA055394(n^7)
%o A274035 (Sage) # Sage cannot handle n = 123, 174, ... without the fallback, even with descent_second_limit = 1000.
%o A274035 def fallback(n):
%o A274035     return gp("my(n=" + str(n) + ");for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0")
%o A274035 def isA055394(z):
%o A274035     z7 = z^7
%o A274035     E = EllipticCurve([0,z7], descent_second_limit = 1000)
%o A274035     try:
%o A274035         for c in E.integral_points():
%o A274035             if c[0] < 0 and c[1] != 0:
%o A274035                 return True
%o A274035         return False
%o A274035     except RuntimeError:
%o A274035         return fallback(z7)
%o A274035 [x for x in range(1, 201) if isA055394(x)]
%Y A274035 Cf. A055394, A174115.
%K A274035 nonn
%O A274035 1,1
%A A274035 _Charles R Greathouse IV_, Jun 06 2016
%E A274035 Missing term 174 inserted by _Jean-François Alcover_, Jan 30 2019