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

2, 5, 8, 9, 10, 12, 15, 17, 24, 26, 28, 31, 33, 36, 37, 40, 43, 44, 46, 50, 52, 54, 56, 57, 63, 65, 68, 69, 72, 73, 76, 80, 82, 89, 91, 98, 100, 101, 108, 113, 122, 126, 127, 128, 129, 134, 136, 141, 145, 148, 150, 152, 161, 164, 168, 170, 171, 174, 177, 183, 185, 189, 192, 196, 197

Offset: 1

Author

Charles R Greathouse IV, Jun 06 2016

Keywords

• nonn

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..3001 (all terms from Charles R Greathouse IV except for a(58)=174)
• Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, Twists of X(7) and primitive solutions to x^2+y^3=z^7, arXiv:math/0508174 [math.NT], 2005; Duke Math. J. 137:1 (2007), pp. 103-158.

Crossrefs

Cf. A055394, A174115.

Programs

Mathematica

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];
Reap[For[n = 1, n < 200, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)

PARI

isA055394(n)=for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0
is(n)=isA055394(n^7)

Sage

# Sage cannot handle n = 123, 174, ... without the fallback, even with descent_second_limit = 1000.
def fallback(n):
    return gp("my(n=" + str(n) + ");for(k=1,sqrtnint(n-1,3),if(issquare(n-k^3),return(1)));0")
def isA055394(z):
    z7 = z^7
    E = EllipticCurve([0,z7], descent_second_limit = 1000)
    try:
        for c in E.integral_points():
            if c[0] < 0 and c[1] != 0:
                return True
        return False
    except RuntimeError:
        return fallback(z7)
[x for x in range(1, 201) if isA055394(x)]

Extensions

Missing term 174 inserted by Jean-François Alcover, Jan 30 2019