A266548 -id:A266548 - cp's OEIS frontend

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.

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}]

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

Zhi-Wei Sun, Jan 02 2016

nonn

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

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

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

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

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

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}]

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

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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.

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