A266004 -id:A266004 - cp's OEIS frontend

A266152 Least positive integer y such that n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

8, 1, 2, 17, 1, 3, 139, 19, 37, 1, 3, 9, 2, 7, 3, 1411, 1, 2, 2, 1, 5, 4, 387, 3, 1, 1, 4, 7, 9, 2, 35, 1, 33, 2, 6, 5, 1, 4, 3, 11, 1, 6, 2, 429, 2, 5, 11, 179, 73, 1, 15, 1, 4, 3, 11, 3, 5, 2, 3, 15, 5, 6, 7, 3, 1, 6, 4, 6337, 8, 16, 3

Offset: 0

Zhi-Wei Sun, Dec 22 2015

nonn

Conjecture: Any integer m can be written as x^4 - y^3 + z^2, where x, y and z are positive integers.
This is slightly stronger than the conjecture in A266003.
See also A266153 for a related sequence, and A266212 for a stronger conjecture.
If n is a positive square, then a(n) = 1. - Altug Alkan, Dec 23 2015

			a(0) = 8 since 0 = 4^4 - 8^3 + 16^2.
a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2.
a(15) = 1411 since 15 = 119^4 - 1411^3 + 51075^2.
a(11019) = 71383 since 11019 = 4325^4 - 71383^3 + 3719409^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Checking the conjecture for integers m with |m| <= 10^5
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, A266153, A266212.

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
Do[y=1;Label[bb];Do[If[SQ[n+y^3-x^4],Print[n," ",y];Goto[aa]],{x,1,(n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Continue,{n,0,70}]

A266153 Least positive integer y such that -n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

Original entry on oeis.org

3, 3, 2, 6, 13, 2, 3, 5, 5, 3, 28, 4, 15, 4, 10, 33, 3, 7, 5, 238, 31, 3, 4, 5, 3, 11, 4, 5, 21, 11, 6, 4, 17, 11, 5, 98, 7, 4, 4, 5, 147, 19, 5, 4, 5, 6, 4, 29, 75, 1011, 7, 9, 7, 4, 8, 6, 59, 47, 4, 5, 71, 4, 17, 45, 13, 7, 18, 9, 175, 8

Offset: 1

Zhi-Wei Sun, Dec 22 2015

nonn

The conjecture in A266152 implies that a(n) > 0 for all n > 0.

It seems that a(n) < n*(n+4)/2 for all n > 1.

			a(1) = 3 since -1 = 1^4 - 3^3 + 5^2.
a(2) = 3 since -2 = 2^4 - 3^3 + 3^2.
a(11) = 28 since -11 = 5^4 - 28^3 + 146^2.
a(20) = 238 since -20 = 32^4 - 238^3 + 3526^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A000583, A262827, A266004, A266152.

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
Do[y=Floor[n^(1/3)]+1;Label[bb];Do[If[SQ[-n+y^3-x^4],Print[n," ",y];Goto[aa]],{x,1,(-n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Continue,{n,1,70}]

A266212 Positive integers x such that x^3 = y^4 + z^2 for some positive integers y and z.

Original entry on oeis.org

8, 13, 20, 25, 40, 125, 128, 193, 200, 208, 225, 313, 320, 328, 400, 500, 605, 640, 648, 1000, 1053, 1156, 1521, 1620, 1625, 1681, 1700, 2000, 2025, 2048, 2125, 2465, 2493, 2873, 2920, 3025, 3088, 3185, 3200, 3240, 3328, 3400, 3600, 3656, 3748, 3816, 4225, 4625, 4913, 5000, 5008, 5120, 5248, 6400, 6728, 6760, 6793, 6845, 7225, 8000

Offset: 1

Zhi-Wei Sun, Dec 23 2015

nonn

If x^3 = y^4 + z^2, then (a^(4k)*x)^3 = (a^(3k)*y)^4 + (a^(6k)*z)^2 for all a = 1,2,3,... and k = 0,1,2,... So the sequence has infinitely many terms.
Conjecture: For any integer m, there are infinitely many triples (x,y,z) of positive integers with x^4 - y^3 + z^2 = m.
This is stronger than the conjecture in A266152.

			a(1) = 8 since 8^3 = 4^4 + 16^2.
a(2) = 13 since 13^3 = 3^4 + 46^2.
a(3) = 20 since 20^3 = 4^4 + 88^2.
a(8) = 193 since 193^3 = 6^4 + 2681^2.
a(12) = 313 since 313^3 = 66^4 + 3419^2.
a(20) = 1000 since 1000^3 = 100^4 + 30000^2.

Zhi-Wei Sun and Chai Wah Wu, Table of n, a(n) for n = 1..698 n = 1..100 from Zhi-Wei Sun
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.

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
n=0;Do[Do[If[SQ[x^3-y^4],n=n+1;Print[n," ",x];Goto[aa]],{y,1,x^(3/4)}];Label[aa];Continue,{x,1,8000}]

A266003 Least nonnegative integer y such that n = x^4 - y^3 + z^2 for some nonnegative integers x and z, or -1 if no such y exists.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 139, 19, 1, 0, 0, 9, 2, 7, 3, 1, 0, 0, 2, 1, 0, 4, 3, 3, 1, 0, 0, 7, 2, 2, 19, 1, 0, 2, 6, 1, 0, 0, 3, 11, 1, 0, 2, 429, 2, 5, 11, 179, 1, 0, 0, 1, 0, 3, 3, 3, 2, 2, 3, 15, 5, 6, 7, 1, 0, 0, 4, 6337, 8, 16, 3, 5, 2, 2, 2, 31, 6, 2, 11, 1, 0, 0, 0, 17, 1, 0, 11, 5, 18, 1, 0, 621, 2, 2, 3, 3, 1, 0, 2, 1, 0

Offset: 0

Zhi-Wei Sun, Dec 19 2015

nonn

Conjecture: Any integer m can be written as x^4 - y^3 + z^2, where x, y and z are nonnegative integers.
I have verified this for all integers m with |m| <= 10^5.
See also A266004 for a related sequence.

			a(6) = 139 since 6 = 36^4 - 139^3 + 1003^2.
a(67) = 6337 since 67 = 676^4 - 6337^3 + 213662^2.
a(176) = 13449 since 176 = 140^4 - 13449^3 + 1559555^2.
a(2667) = 661^4 - 15655^3 + 1909401^2.
a(11019) = 71383 since 11019 = 4325^4 - 71383^3 + 3719409^2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Checking the conjecture for m = 0..10^5

Cf. A000290, A000578, A000583, A262827, A266004.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[y=0;Label[bb];Do[If[SQ[n+y^3-x^4],Goto[aa]],{x,0,(n+y^3)^(1/4)}];y=y+1;Goto[bb];Label[aa];Print[n," ",y];Continue,{n,0,100}]

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

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

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A266152 Least positive integer y such that n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

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A266153 Least positive integer y such that -n = x^4 - y^3 + z^2 for some positive integers x and z, or 0 if no such y exists.

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A266212 Positive integers x such that x^3 = y^4 + z^2 for some positive integers y and z.

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A266003 Least nonnegative integer y such that n = x^4 - y^3 + z^2 for some nonnegative integers x and z, or -1 if no such y exists.

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

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