A266152 -id:A266152 - cp's OEIS frontend

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.

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}]

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}]

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}]

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}]

A266314 Least positive integer x such that n + x^7 = y^2 + z^2 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

2, 1, 2, 13, 1, 3, 7, 1, 2, 1, 3, 15, 1, 6, 11, 11, 1, 1, 2, 1, 2, 2, 7, 3, 1, 1, 3, 5, 1, 2, 7, 1, 2, 1, 2, 5, 1, 4, 3, 1, 1, 2, 2, 7, 1, 2, 7, 3, 5, 1, 2, 1, 1, 2, 11, 21, 5, 1, 3, 5, 1, 3, 3, 3, 1, 2, 2, 1, 4, 2, 3

Offset: 0

Zhi-Wei Sun, Dec 27 2015

nonn

The general conjecture in A266277 implies that for each odd prime p and any integer m there are positive integers x, y and z such that m + x^p = y^2 + z^2.
For k = 4,6,8,... and any integer m == 6 (mod 8), there are no integers x, y and z with m + x^k = y^2 + z^2 since m + x^k with x an integer is congruent to 6 or 7 modulo 8.
As 2j+1 = (j+1)^2 - j^2, if m - z^k is odd with |m - z^k| > 1 then m + x^2 = y^2 + z^k for some positive integers x and y.

			a(2) = 2 since 2 + 2^7 = 3^2 + 11^2.
a(3) = 13 since 3 + 13^7 = 554^2 + 7902^2.
a(5) = 3 since 5 + 3^7 = 16^2 + 44^2.
a(6) = 7 since 6 + 7^7 = 30^2 + 907^2.
a(462) = 71 since 462 + 71^7 = 456497^2 + 2981062^2.

Zhi-Wei Sun and Chai Wah Wu, Table of n, a(n) for n = 0..10000 n = 0..2000 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. Section 5.)

Cf. A000290, A001015, A266152, A266153, A266230, A266231, A266277.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[x=1;Label[bb];Do[If[SQ[n+x^7-y^2],Print[n," ",x];Goto[aa]],{y,1,Sqrt[(n+x^7)/2]}];x=x+1;Goto[bb];Label[aa];Continue,{n,1,70}]
(* second program: *)
xmax = 100; r[n_, x_] := Reduce[y>0 && z>0 && n+x^7 == y^2+z^2, {y, z}, Integers]; a[n_] := For[x=1, x <= xmax, x++, If[r[n, x] =!= False, Return[x]]] /. Null -> 0; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 27 2015 *)

A266363 Least positive integer x such that n + x^2 = y^3 + z^4 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

3, 1, 302, 5, 47, 2, 362, 6, 1, 372, 14, 61, 4, 2, 70, 3, 1, 24, 5, 3, 2, 14, 364, 1, 2, 8, 10, 1, 454, 6, 848, 7, 15, 7, 3, 18, 14, 13, 1362, 2, 5, 10, 1, 37, 6, 9, 6, 68, 13, 4, 24, 36, 37, 6, 26, 5, 3, 5, 15, 7, 9

Offset: 0

Zhi-Wei Sun, Dec 28 2015

nonn

The general conjecture in A266277 implies that for any integer m there are positive integers x, y and z such that m + x^2 = y^3 + z^4.

See also A266152 and A266364 for similar sequences.

			a(0) = 3 since 0 + 3^2 = 2^3 + 1^4.
a(2) = 302 since 2 + 302^2 = 45^3 + 3^4.
a(3) = 5 since 3 + 5^2 = 3^3 + 1^4.
a(38) = 1362 since 38 + 1362^2 = 121^3 + 17^4.
a(394) = 110307 since 394 + 110307^2 = 2283^3 + 128^4.
a(5546) = 945840 since 5546 + 945840^2 = 9625^3 + 233^4.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000290, A000578, A000583, A266152, A266153, A266230, A266231, A266277, A266314, A266364.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^4],Print[n," ",x];Goto[aa]],{y,1,(n+x^2-1)^(1/4)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,60}]

A266364 Least positive integer x such that n + x^4 = y^2 + z^3 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

6, 1, 69, 7, 1, 46, 13, 5, 1, 1, 2, 1, 2, 4, 27, 2, 1, 2, 28, 3, 2, 2, 37, 1, 4, 1, 11, 1, 2, 5, 1, 5, 1, 4, 2, 1, 1, 8, 4, 6, 8, 2, 1, 1, 6, 3, 3, 2, 3, 1, 18, 1, 2, 3, 6, 9, 1, 2, 6, 5, 2

Offset: 0

Zhi-Wei Sun, Dec 28 2015

nonn

The general conjecture in A266277 implies that for any integer m there are positive integers x, y and z such that m + x^2 = y^3 + z^4.

See also A266152 and A266363 for similar sequences.

			a(0) = 6 since 0 + 6^4 = 28^2 + 8^3.
a(2) = 69 since 2 + 69^4 = 44^2 + 283^3.
a(5) = 46 since 5 + 46^4 = 1742^2 + 113^3.
a(570) = 983 since 570 + 983^4 = 546596^2 + 8595^3.
a(8078) = 2255 since 8078 + 2255^4 = 1926054^2 + 28083^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Cf. A000290, A000578, A000583, A266152, A266153, A266230, A266231, A266277, A266314, A266363.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[x=1;Label[bb];Do[If[SQ[n+x^4-y^3],Print[n," ",x];Goto[aa]],{y,1,(n+x^4-1)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,60}]

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}]

A266548 Least prime p such that n + p^3 = x^2 + y^3 for some positive integers x and y, or 0 if no such prime p exists.

Original entry on oeis.org

71, 2, 2, 5, 2, 3767, 3, 7, 7, 2, 3, 23, 53, 13, 17, 13, 2, 3, 2, 7, 2, 23, 11, 2, 17, 2, 7, 5, 2, 2, 3, 19, 257, 8039, 13, 2, 2, 59, 3, 5, 17, 3, 2, 61, 2, 3, 3, 37, 313, 2, 631, 17, 5, 3, 17, 2, 17, 2, 7, 97, 2, 47, 3, 29, 2, 2, 31, 47, 2, 7, 19

Offset: 0

Zhi-Wei Sun, Dec 31 2015

nonn

Conjecture: (i) Any integer can be written as x^2 + y^3 - p^3, where x and y are positive integers, and p is a prime.
(ii) Each integer can be written as x^2 - y^3 + p^3, where x and y are positive integers, and p is a prime.
See also A266230 and A266277 for related conjectures.
Is every prime in this sequence? - David A. Corneth, Dec 30 2017

			a(0) = 71 since 0 + 71^3 = 588^2 + 23^3 with 71 prime.
a(3) = 5 since 3 + 5^3 = 8^2 + 4^3 with 5 prime.
a(5) = 3767 since 5 + 3767^3 = 214886^2 + 1938^3 with 3767 prime.
a(2966) = 68371 since 2966 + 68371^3 = 17867983^2 + 6992^3 with 68371 prime.
a(7880) = 51137 since 7880 + 51137^3 = 10176509^2 + 31128^3 with 51137 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000040, A000290, A000578, A055394, A266152, A266153, A266230, A266231, A266277, A266314, A266363, A266364, A266528.

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

isokp(p, n) = {my(s = n+p^3); for (k=1, sqrtnint(s, 3), if ((q=s-k^3) && issquare(q), return (1)););}
a(n) = {p = 2; while(!isokp(p, n), p = nextprime(p+1)); p;} \\ Michel Marcus, Jan 04 2016

a(n, {plim = 2}) = forprime(p = plim, oo, c = n + p^3; for(i = 1, sqrtnint(c, 3), if(issquare(c - i^3) && c - i^3 > 0, return(p))))
first(n, {plim = 100}) = {my(res = vector(n), l = List(), s, i, c);
for(u=1, sqrtint((n+plim^3)\1-1), for(v=1, sqrtnint((n+plim^3)\1-u^2, 3), listput(l, u^2+v^3))); l = Set(l); forprime(p = 2, plim, s = 1; while(l[s] < p^3 + 1, s++); for(i = s, #l, c = l[i] - p^3; if(c <= n, if(res[c] == 0, res[c] = p)
, next(2)))); for(i = 1, n, if(res[i] == 0, res[i] = a(i, plim + 1))); concat([71], res)} \\ David A. Corneth, Dec 30 2017

cp's OEIS Frontend

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

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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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A266314 Least positive integer x such that n + x^7 = y^2 + z^2 for some positive integers y and z, or 0 if no such x exists.

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A266363 Least positive integer x such that n + x^2 = y^3 + z^4 for some positive integers y and z, or 0 if no such x exists.

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A266364 Least positive integer x such that n + x^4 = y^2 + z^3 for some positive integers y and z, or 0 if no such x exists.

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