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

0,1

Comments

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

Links

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.)

Maple

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.

Mathematica

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}]

Crossrefs

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

Sequence in context: A344050 A213069 A357813 * A016482 A012854 A370331

Adjacent sequences: A266274 A266275 A266276 * A266278 A266279 A266280

Keyword

nonn

Author

Zhi-Wei Sun, Dec 26 2015

Status

approved