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%I #21 Jun 06 2018 11:36:19
%S 0,1,1,1,1,2,1,3,2,2,2,4,3,2,2,3,3,3,2,2,2,4,3,2,1,5,4,3,2,5,5,5,5,3,
%T 3,5,5,4,4,4,5,5,2,5,3,5,4,7,2,4,6,6,5,4,4,5,8,4,4,4,7,6,4,3,4,8,4,7,
%U 3,3,6,8,2,5,6,5,4,6,4,3
%N Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 5^c + 5^d, where a,b,c,d are nonnegative integers with a <= b and c <= d.
%C Conjecture: a(n) > 0 for all n > 1. In other words, any integers n > 1 can be written as the sum of two triangular numbers and two powers of 5.
%C This has been verified for all n = 2..10^10.
%C See A303393 for the numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers.
%C See also A303401, A303432 and A303540 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A303389/b303389.txt">Table of n, a(n) for n = 1..100000</a>
%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.
%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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(4) = 1 with 4 = 1*(1+1)/2 + 1*(1+1)/2 + 5^0 + 5^0.
%e a(5) = 1 with 5 = 0*(0+1)/2 + 2*(2+1)/2 + 5^0 + 5^0.
%e a(7) = 1 with 7 = 0*(0+1)/2 + 1*(1+1)/2 + 5^0 + 5^1.
%e a(25) = 1 with 25 = 0*(0+1)/2 + 5*(5+1)/2 + 5^1 + 5^1.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t f[n_]:=f[n]=FactorInteger[n];
%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
%t QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
%t tab={};Do[r=0;Do[If[QQ[4(n-5^j-5^k)+1],Do[If[SQ[8(n-5^j-5^k-x(x+1)/2)+1],r=r+1],{x,0,(Sqrt[4(n-5^j-5^k)+1]-1)/2}]],{j,0,Log[5,n/2]},{k,j,Log[5,n-5^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000217, A000351, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303363, A303393, A303401, A303432, A303540.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Apr 23 2018
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