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A343528 Number of ways to write n as x^4 + T(y)^2 + T(z) + 2^w, where x,y,z are nonnegative integers, w is a positive integer, and T(m) denotes the triangular number m*(m+1)/2.

%I A343528 #15 Apr 21 2021 04:29:42
%S A343528 0,1,3,4,5,5,3,4,6,5,5,6,5,6,6,4,7,10,10,9,7,4,7,10,7,8,9,7,5,7,7,10,
%T A343528 13,9,8,7,5,8,14,9,10,11,6,9,10,8,10,13,8,7,6,5,11,15,9,7,8,6,8,10,10,
%U A343528 10,10,6,7,9,6,10,17,10,9,9,6,10,10,6,9,9,6,10,9,6,11,14,8,11,11,9,11,11
%N A343528 Number of ways to write n as x^4 + T(y)^2 + T(z) + 2^w, where x,y,z are nonnegative integers, w is a positive integer, and T(m) denotes the triangular number m*(m+1)/2.
%C A343528 Conjecture: a(n) > 0 for all n > 1.
%C A343528 We have verified a(n) > 0 for all 1 < n <= 2*10^7.
%C A343528 Conjecture verified up to 10^10. - _Giovanni Resta_, Apr 21 2021
%H A343528 Zhi-Wei Sun, <a href="/A343528/b343528.txt">Table of n, a(n) for n = 1..10000</a>
%e A343528 a(2) = 1 with 2 = 0^4 + T(0)^2 + T(0) + 2^1.
%e A343528 a(7) = 3, and 7 = 0^4 + T(0)^2 + T(2) + 2^2 = 1^4 + T(1)^2 + T(1) + 2^2 = 1^4 + T(1)^2 + T(2) + 2^1.
%t A343528 TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
%t A343528 tab={};Do[r=0;Do[If[TQ[n-x^4-(y(y+1)/2)^2-2^k],r=r+1],{x,0,(n-1)^(1/4)},{y,0,(Sqrt[8*Sqrt[n-x^4-1]+1]-1)/2},{k,1,Log[2,n-x^4-(y(y+1)/2)^2]}];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y A343528 Cf. A000079, A000217, A000290, A000583, A343397, A343411, A343460.
%K A343528 nonn
%O A343528 1,3
%A A343528 _Zhi-Wei Sun_, Apr 18 2021