|
%I #30 Jul 19 2023 19:52:10
%S 1,2,1,2,4,2,1,2,2,2,1,1,3,3,1,2,5,3,1,4,4,2,2,1,2,3,1,4,8,4,1,4,4,1,
%T 1,5,8,5,3,3,3,2,1,6,6,1,1,4,7,5,3,8,10,5,2,1,3,3,2,5,5,2,3,8,8,4,2,7,
%U 8,1,1,1,3,3,2,7,7,4,3,6
%N Number of ordered ways to write n as w^2 + 3*x^2 + y^4 + z^5, where w is a positive integer and x,y,z are nonnegative integers.
%C Conjectures:
%C (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 7, 11, 12, 15, 19, 24, 27, 31, 34, 35, 43, 46, 47, 56, 70, 71, 72, 87, 88, 115, 136, 137, 147, 167, 168, 178, 207, 235, 236, 267, 286, 297, 423, 537, 747, 762, 1017.
%C (ii) Any positive integer n can be written as w^2 + x^4 + y^5 + pen(z), where w is a positive integer, x,y,z are nonnegative integers, and pen(z) denotes the pentagonal number z*(3*z-1)/2.
%C Conjectures a(n) > 0 and (ii) verified up to 10^11. - _Mauro Fiorentini_, Jul 19 2023
%C See also A262813, A262857, A270566, A271106 and A271325 for some other conjectures on representations.
%H Zhi-Wei Sun, <a href="/A273917/b273917.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.NT], 2016-2017.
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190. (See Remark 1.1.)
%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. (See Conjecture 3.2.)
%e a(1) = 1 since 1 = 1^2 + 3*0^2 + 0^4 + 0^5.
%e a(3) = 1 since 3 = 1^2 + 3*0^2 + 1^4 + 1^5.
%e a(7) = 1 since 7 = 2^2 + 3*1^2 + 0^4 + 0^5.
%e a(11) = 1 since 11 = 3^2 + 3*0^2 + 1^4 + 1^5.
%e a(12) = 1 since 12 = 3^2 + 3*1^2 + 0^4 + 0^5.
%e a(15) = 1 since 15 = 1^2 + 3*2^2 + 1^4 + 1^5.
%e a(19) = 1 since 19 = 4^2 + 3*1^2 + 0^4 + 0^5.
%e a(24) = 1 since 24 = 2^2 + 3*1^2 + 2^4 + 1^5.
%e a(27) = 1 since 27 = 5^2 + 3*0^2 + 1^4 + 1^5.
%e a(31) = 1 since 31 = 2^2 + 3*3^2 + 0^4 + 0^5.
%e a(34) = 1 since 34 = 1^2 + 3*0^2 + 1^4 + 2^5.
%e a(35) = 1 since 35 = 4^2 + 3*1^2 + 2^4 + 0^5.
%e a(43) = 1 since 43 = 4^2 + 3*3^2 + 0^4 + 0^5.
%e a(46) = 1 since 46 = 1^2 + 3*2^2 + 1^4 + 2^5.
%e a(47) = 1 since 47 = 2^2 + 3*3^2 + 2^4 + 0^5.
%e a(56) = 1 since 56 = 6^2 + 3*1^2 + 2^4 + 1^5.
%e a(70) = 1 since 70 = 5^2 + 3*2^2 + 1^4 + 2^5.
%e a(71) = 1 since 71 = 6^2 + 3*1^2 + 0^4 + 2^5.
%e a(72) = 1 since 72 = 6^2 + 3*1^2 + 1^4 + 2^5.
%e a(87) = 1 since 87 = 6^2 + 2*1^2 + 2^4 + 2^5.
%e a(88) = 1 since 88 = 2^2 + 3*1^2 + 3^4 + 0^5.
%e a(115) = 1 since 115 = 8^2 + 3*1^2 + 2^4 + 2^5.
%e a(136) = 1 since 136 = 10^2 + 3*1^2 + 1^4 + 2^5.
%e a(137) = 1 since 137 = 11^2 + 3*0^2 + 2^4 + 0^5.
%e a(147) = 1 since 147 = 12^2 + 3*1^2 + 0^4 + 0^5.
%e a(167) = 1 since 167 = 2^2 + 3*7^2 + 2^4 + 0^5.
%e a(168) = 1 since 168 = 2^2 + 3*7^2 + 2^4 + 1^5.
%e a(178) = 1 since 178 = 7^2 + 3*4^2 + 3^4 + 0^5.
%e a(207) = 1 since 207 = 10^2 + 3*5^2 + 0^4 + 2^5.
%e a(235) = 1 since 235 = 12^2 + 3*5^2 + 2^4 + 0^5.
%e a(236) = 1 since 236 = 12^2 + 3*5^2 + 2^4 + 1^5.
%e a(267) = 1 since 267 = 12^2 + 3*5^2 + 2^4 + 2^5.
%e a(286) = 1 since 286 = 4^2 + 3*3^2 + 0^4 + 3^5.
%e a(297) = 1 since 297 = 3^2 + 3*0^2 + 4^4 + 2^5.
%e a(423) = 1 since 423 = 11^2 + 3*10^2 + 1^4 + 1^5.
%e a(537) = 1 since 537 = 21^2 + 3*4^2 + 2^4 + 2^5.
%e a(747) = 1 since 747 = 11^2 + 3*0^2 + 5^4 + 1^5.
%e a(762) = 1 since 762 = 27^2 + 3*0^2 + 1^4 + 2^5.
%e a(1017) = 1 since 1017 = 27^2 + 3*0^2 + 4^4 + 2^5.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-3*x^2-y^4-z^5],r=r+1],{x,0,Sqrt[(n-1)/3]},{y,0,(n-1-3x^2)^(1/4)},{z,0,(n-1-3x^2-y^4)^(1/5)}];Print[n," ",r];Continue,{n,1,80}]
%Y Cf. A000118, A000290, A000326, A000583, A000584, A262813, A262827, A262857, A270566, A270969, A271076, A271106, A273429, A273915.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jun 04 2016
|
