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%I #27 Apr 08 2020 08:44:54
%S 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,
%T 10,1,501,2,15205,3,1,88,5,44,2,60,2537,1,5,52,32834,4,18,9,84,7,13,4,
%U 3,16,14,39,26,2,3,10,1,20,6,2,8,543,1,111,4570,36,110,1402,501
%N 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.
%C 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.
%C This implies that a(n) > 0 for all n. Also, it includes the conjectures in A266152, A266212 and A266230 as special cases.
%C 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
%H Zhi-Wei Sun, <a href="/A266277/b266277.txt">Table of n, a(n) for n = 0..2315</a>
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120. (Cf. Section 5.)
%p a(0) = 3 since 0 + 3^2 = 2^3 + 1^5.
%p a(2) = 83 since 2 + 83^2 = 19^3 + 2^5.
%p a(42) = 32834 since 42 + 32834^2 = 781^3 + 57^5.
%p a(445) = 903402 since 445 + 903402^2 = 9345^3 + 34^5.
%p a(510) = 10875037 since 510 + 10875037^2 = 40712^3 + 551^5.
%t CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
%t 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}]
%Y Cf. A000290, A000578, A000584, A266152, A266153, A266212, A266215, A266230, A266231, A266314, A266363, A266364, A266528, A266548, A266651, A266985.
%K nonn
%O 0,1
%A _Zhi-Wei Sun_, Dec 26 2015
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