cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Dec 24 2015

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • 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}]

Extensions

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

A266651 Nonnegative integers x such that x^3 + 6^3 is a sum of two squares.

Original entry on oeis.org

14, 21, 62, 190, 206, 210, 237, 286, 334, 350, 382, 398, 426, 430, 446, 453, 574, 622, 670, 734, 766, 777, 782, 878, 958, 974, 1102, 1294, 1317, 1342, 1438, 1486, 1678, 1694, 1722, 1749, 1774, 1790, 1938, 1965, 1966, 2014, 2030, 2110, 2126, 2154, 2222, 2254, 2270, 2289, 2302, 2397, 2414, 2446, 2558, 2638, 2686, 2721, 2734, 2750
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 02 2016

Keywords

Comments

Conjecture: For any integer x with gcd(x,6) = 1, the number x^3 + 6^3 is never a sum of two squares.
We have verified this for x up to 5*10^6.
Note also that 6^3 + (-2)^3 = 8^2 + 12^2.
Hao Pan at Nanjing Univ. confirmed the conjecture on Jan. 3, 2016. - Zhi-Wei Sun, Jan 06 2016

Examples

			a(1) = 14 since 14^3 + 6^3 = 16^2 + 52^2.
a(7) = 237 since 237^3 + 6^3 = 162^2 + 3645^2.

Links

Crossrefs

Cf. A000290, A000578, A001481, A266152, A266230, A266231, A266277, A266363, A266364, A266548.

Programs

  • Mathematica
    f[n_]:=f[n]=FactorInteger[n]
Le[n_]:=Le[n]=Length[f[n]]
n=0;Do[Do[If[Mod[Part[Part[f[x^3+6^3],i],1],4]==3&&Mod[Part[Part[f[x^3+6^3],i],2],2]==1,Goto[aa]],{i,1,Le[216+x^3]}];n=n+1;Print[n," ",x];Label[aa];Continue,{x,0,2750}]

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

Views

Author

Zhi-Wei Sun, Jan 08 2016

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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}]
Previous Showing 11-13 of 13 results.

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