# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a303389 Showing 1-1 of 1 %I A303389 #21 Jun 06 2018 11:36:19 %S A303389 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 A303389 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 A303389 3,3,6,8,2,5,6,5,4,6,4,3 %N A303389 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 A303389 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 A303389 This has been verified for all n = 2..10^10. %C A303389 See A303393 for the numbers of the form x*(x+1)/2 + 5^y with x and y nonnegative integers. %C A303389 See also A303401, A303432 and A303540 for similar conjectures. %H A303389 Zhi-Wei Sun, Table of n, a(n) for n = 1..100000 %H A303389 Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. %H A303389 Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120. %H A303389 Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018. %e A303389 a(4) = 1 with 4 = 1*(1+1)/2 + 1*(1+1)/2 + 5^0 + 5^0. %e A303389 a(5) = 1 with 5 = 0*(0+1)/2 + 2*(2+1)/2 + 5^0 + 5^0. %e A303389 a(7) = 1 with 7 = 0*(0+1)/2 + 1*(1+1)/2 + 5^0 + 5^1. %e A303389 a(25) = 1 with 25 = 0*(0+1)/2 + 5*(5+1)/2 + 5^1 + 5^1. %t A303389 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]; %t A303389 f[n_]:=f[n]=FactorInteger[n]; %t A303389 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 A303389 QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]); %t A303389 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 A303389 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 A303389 nonn %O A303389 1,6 %A A303389 _Zhi-Wei Sun_, Apr 23 2018 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE