A266212 -id:A266212 - 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}]

A266230 Least positive integer x such that n + x^2 = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

3, 1, 3703, 5, 43, 2, 119, 3, 1, 19, 5, 384, 2, 29, 29, 1, 7, 18, 6, 3, 13, 14, 869, 7, 2, 15, 3, 1, 10, 5, 23, 2, 20, 10, 1, 45, 6, 2373, 4, 1193, 5, 52, 7, 36, 54, 3, 18, 5, 13, 4, 2, 385, 9, 1, 14, 6, 3, 76, 250, 250, 34, 2, 8, 3, 1, 336, 5, 52, 2, 8, 28, 1, 21, 12, 13, 4, 113

Offset: 0

Zhi-Wei Sun, Dec 24 2015

nonn

Conjecture:  For any integer m, there are positive integers x, y and z such that m + x^2 = y^3 + z^3.
This is similar to the conjecture in A266152. We have verified it for all integers m with |m| <= 25000.
Obviously, a(k^3) = 1 for any positive integer k.
See also A266231 for a related sequence.

			a(0) = 3 since 0 + 3^2 = 1^3 + 2^3.
a(2) = 3703 since 2 + 3703^2 = 107^3 + 232^3.
a(3) = 5 since 3 + 5^2 = 1^3 + 3^3.
a(4) = 43 since 4 + 43^2 = 5^3 + 12^3.
a(37) = 2373 since 37 + 2373^2 = 93^3 + 169^3.
a(1227) = 132316 since 1227 + 132316^2 = 1874^3 + 2219^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Checking the conjecture for integers m with 10000 < |m| <= 25000

Cf. A000290, A000578, A003325, A266152, A266153, A266212, A266231.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^3],Print[n," ",x];Goto[aa]],{y,1,((n+x^2)/2)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,80}]

A266231 Least positive integer x such that x^2 - n = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

6, 2, 61, 47, 3283, 16, 3, 6, 5, 8, 12, 686, 16, 4, 302, 5, 13, 12, 152, 6, 7, 83, 5, 148, 33, 37, 6, 10, 8, 11, 34, 16, 7, 6, 10, 8, 24, 53, 16, 7, 13, 52, 13, 14, 30, 9, 7, 8, 11, 67, 74, 22, 9, 28, 8, 11, 43, 115, 20, 122, 23, 8, 14, 48, 9, 25, 11, 14, 392, 14

Offset: 1

Zhi-Wei Sun, Dec 24 2015

nonn

The conjecture in A266230 implies that a(n) > 0 for all n > 0.

			 a(1) = 6 since 6^2 - 1 = 2^3 + 3^3.
a(3) = 61 since 61^2 - 3 = 7^3 + 15^3.
a(4) = 47 since 47^2 - 4 = 2^3 + 13^3.
a(5) = 3283 since 3283^2 - 5 = 65^3 + 219^3.
a(166) = 6554 since 6554^2 - 166 = 175^3 + 335^3.
a(635) = 44779 since 44779^2 - 635 = 25^3 + 1261^3.

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

Cf. A000290, A000578, A003325, A266152, A266153, A266212, A266230.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=Floor[Sqrt[n]]+1;Label[bb];Do[If[CQ[-n+x^2-y^3],Print[n," ",x];Goto[aa]],{y,1,((-n+x^2)/2)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,1,70}]

A266277 Least positive integer x such that n + x^2 = y^3 + z^5 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

3, 1, 83, 5, 6, 2, 175, 19, 1, 191, 7, 31, 4, 12, 16, 5, 7, 4, 17, 3, 18, 14, 1099, 6, 2, 244, 10, 1, 501, 2, 15205, 3, 1, 88, 5, 44, 2, 60, 2537, 1, 5, 52, 32834, 4, 18, 9, 84, 7, 13, 4, 3, 16, 14, 39, 26, 2, 3, 10, 1, 20, 6, 2, 8, 543, 1, 111, 4570, 36, 110, 1402, 501

Offset: 0

Zhi-Wei Sun, Dec 26 2015

nonn

Conjecture: If {a,b,c} is among the multisets {2,2,p} (p is an odd prime or a product of primes congruent to 1 modulo 4) and {2,3,k} (k = 3,4,5), then for any integer m there are (infinitely many) triples (x,y,z) of positive integers such that m = x^a + y^b - z^c.
This implies that a(n) > 0 for all n. Also, it includes the conjectures in A266152, A266212 and A266230 as special cases.
For any odd prime p == 3 (mod 4) and odd integer n > 1, I have proved that x^{pn} + (2p)^p with x an integer is never a sum of two squares.  - Zhi-Wei Sun, Jan 06 2016

Zhi-Wei Sun, Table of n, a(n) for n = 0..2315
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120. (Cf. Section 5.)

Cf. A000290, A000578, A000584, A266152, A266153, A266212, A266215, A266230, A266231, A266314, A266363, A266364, A266528, A266548, A266651, A266985.

a(0) = 3 since 0 + 3^2 = 2^3 + 1^5.
a(2) = 83 since 2 + 83^2 = 19^3 + 2^5.
a(42) = 32834 since 42 + 32834^2 = 781^3 + 57^5.
a(445) = 903402 since 445 + 903402^2 = 9345^3 + 34^5.
a(510) = 10875037 since 510 + 10875037^2 = 40712^3 + 551^5.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^5],Print[n," ",x];Goto[aa]],{y,1,(n+x^2-1)^(1/5)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,70}]

A266528 Least positive integer x such that n + x^5 = y^2 + z^3 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

8, 1, 8, 3, 1, 2, 11, 5, 1, 1, 42, 1, 2, 11, 3, 21, 1, 3, 2, 5, 2, 3, 3, 1, 7, 1, 3, 1, 22, 4, 1, 2, 1, 2, 8, 1, 1, 3, 5, 13, 2, 2, 1, 1, 2, 27, 3, 3, 2, 1, 2, 1, 7, 6, 3, 5, 1, 2, 7, 2, 5, 15, 1, 17, 1, 13, 4, 1, 2, 2, 86

Offset: 0

Zhi-Wei Sun, Dec 31 2015

nonn

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

			a(0) = 8 since 0 + 8^5 = 104^2 + 28^3.
a(2) = 8 since 2 + 8^5 = 179^2 + 9^3.
a(6) = 11 since 6 + 11^5 = 143^2 + 52^3.
a(10) = 42 since 10 + 42^5 = 11415^2 + 73^3.
a(15) = 21 since 15 + 21^5 = 1355^2 + 131^3.
a(435) = 3019 since 435 + 3019^5 = 475594653^2 + 290845^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..1100

Cf. A000290, A000578, A000584, A266152, A266153, A266212, A266215, A266230, A266231, A266277, A266314, A266363, A266364.

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

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

A273908 Integers z such that x^2 + y^4 = z^6 where x, y, z > 0, is soluble.

Original entry on oeis.org

5, 15, 20, 34, 39, 41, 45, 55, 60, 65, 80, 85, 111, 125, 135, 136, 145, 150, 156, 164, 175, 180, 194, 219, 220, 240, 245, 255, 260, 265, 299, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 495, 500, 505, 514, 525, 540, 544

Offset: 1

Altug Alkan, Jun 03 2016

nonn

A271576 is a subsequence.
Terms that are not in A271576 are 55, 220, 299, ...
Sequence is infinite since if k is a term then also t^2*k is a term, for every t>0. - Giovanni Resta, Jun 04 2016

			5 is a term because 75^2 + 10^4 = 5^6.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A111925, A266212, A271576.

q[n_] := {} != Select[Range[n^(1/4)]^4, n > # && IntegerQ@ Sqrt[n - #] &]; Select[ Range[100], q[#^6] &] (* Giovanni Resta, Jun 04 2016 *)

A274012 Integers n such that n^3 is the average of a nonzero square and a nonzero fourth power.

Original entry on oeis.org

1, 5, 16, 25, 26, 40, 41, 50, 80, 81, 125, 250, 256, 365, 386, 400, 405, 416, 425, 450, 457, 477, 625, 626, 640, 656, 800, 841, 845, 1000, 1125, 1153, 1210, 1225, 1280, 1296, 1681, 1825, 2000, 2025, 2057, 2106, 2197, 2312, 2401, 3042, 3125, 3240, 3250, 3321, 3362, 3400, 3625

Offset: 1

Altug Alkan, Jun 06 2016

nonn

Numbers n such that 2*n^3 = x^2 + y^4 where x and y are nonzero integers, is soluble.
Square terms of this sequence are 1, 16, 25, 81, 256, 400, 625, 841, 1225, 1296, 1681, 2025, 2401, ...
From David A. Corneth, Jun 06 2016 (Start):
A000351, the powers of 5, is a subsequence.
If n is a term, then n * k^4 is a term; as 2*n^3 = x^4 + y^2, 2 * (n * k^4)^3 = (k^3 * x)^4 + (k^6 * y)^2. (End)

			5 is a term because 5^3 = (13^2 + 3^4) / 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A266212.

is(n) = for(x=1, (2*n) ^ 0.75, if(issquare(2*n^3 - x^4)&&2*n^3-x^4>0, return(1)); 0) \\ David A. Corneth, Jun 06 2016

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.

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A266230 Least positive integer x such that n + x^2 = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

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A266231 Least positive integer x such that x^2 - n = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

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A266277 Least positive integer x such that n + x^2 = y^3 + z^5 for some positive integers y and z, or 0 if no such x exists.

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A266528 Least positive integer x such that n + x^5 = y^2 + z^3 for some positive integers y and z, or 0 if no such x 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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A273908 Integers z such that x^2 + y^4 = z^6 where x, y, z > 0, is soluble.

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A274012 Integers n such that n^3 is the average of a nonzero square and a nonzero fourth power.

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