A266215 Positive integers x such that x^3 - 1 = y^4 + z^2 for some positive integers y and z.

3, 13, 27, 147, 203, 5507, 15661, 16957, 21531, 29931, 38051, 47171, 57147, 84027, 85547, 90891, 167051, 273651, 337501, 392881, 421715, 566691, 609971, 698113, 914701, 1229283, 1435213, 1564573, 1786587, 1987571, 2523387, 2579377, 2716443, 3760347, 3778273

Offset

1,1

Comments

The conjecture in A266212 implies that this sequence has infinitely many terms.

Links

Table of n, a(n) for n=1..35.

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Example

a(1) = 3 since 3^3 - 1 = 1^4 + 5^2.
a(2) = 13 since 13^3 - 1 = 6^4 + 30^2.
a(6) = 5507 since 5507^3 - 1 = 29^4 + 408669^2.
a(16) = 90891 since 90891^3 - 1 = 949^4 + 27387137^2.
a(35) = 3778273 since 3778273^3 - 1 = 85386^4 + 883654380^2.

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
n=0; Do[Do[If[SQ[x^3-1-y^4], n=n+1; Print[n, " ", x]; Goto[aa]], {y, 1, (x^3-1)^(1/4)}]; Label[aa]; Continue, {x, 1, 10^5}]

Crossrefs

Cf. A000290, A000578, A000583, A262827, A266003, A266004, A266152, A266153, A266212.

Sequence in context: A099062 A318368 A196014 * A192535 A002304 A117516

Adjacent sequences: A266212 A266213 A266214 * A266216 A266217 A266218

Keyword

nonn

Author

Zhi-Wei Sun, Dec 24 2015

Extensions

a(17)-a(35) from Lars Blomberg, Dec 30 2015

Status

approved