A266215 -id:A266215 - 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}]

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}]

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.

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