A266153 -id:A266153 - cp's OEIS frontend

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

Zhi-Wei Sun, Dec 24 2015

nonn

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

			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.

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

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

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}]

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

A266985 Least positive integer x such that n + x^3 = y^2 + z^5 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

7, 1, 2, 34, 1, 55, 3, 5, 30, 1, 3, 242, 6, 7, 3, 26, 1, 4, 2, 7, 5, 3, 62, 3, 77, 1, 107, 10, 2, 2, 3, 6, 1, 2, 128, 1, 1, 4, 3, 11, 1, 3, 2, 6, 7, 5, 22, 1, 50, 1, 7, 5, 6, 16, 3, 3, 1, 2, 4, 62, 2, 17, 19, 6, 1, 8, 14, 1, 4, 3, 11

Offset: 0

Zhi-Wei Sun, Jan 08 2016

nonn

The general conjecture in A266277 implies that for any integer m there are positive integers x, y and z with m + x^3 = y^2 + z^5.

See also A266277 and A266528 for similar conjectures.

			a(0) = 7 since 0 + 7^3 = 10^2 + 3^5.
a(3) = 34 since 3 + 34^3 = 150^2 + 7^5.
a(8) = 30 since 8 + 30^3 = 101^2 + 7^5.
a(11) = 242 since 11 + 242^3 = 3420^2 + 19^5.
a(766) = 90891 since 766 + 90891^3 = 11850281^2 + 906^5.

Zhi-Wei Sun, Table of n, a(n) for n = 0..3800

Cf. A000290, A000578, A000584, A266152, A266153, A266230, A266231, A266314, A266363, A266364, A266528, A266548.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[x=1;Label[bb];Do[If[SQ[n+x^3-y^5],Print[n," ",x];Goto[aa]],{y,1,(n+x^3-1)^(1/5)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,70}]

cp's OEIS Frontend

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

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