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A262985 Number of ordered ways to write n as 2^x + phi(y^2) + z*(z+1)/2 with x, y and z positive integers, where phi(.) is Euler's totient function given by A000010.

%I A262985 #15 Jan 01 2023 12:36:36
%S A262985 0,0,0,1,1,2,2,1,3,2,5,2,5,2,5,4,4,4,5,7,3,3,5,5,8,4,5,3,5,4,8,4,3,6,
%T A262985 5,2,9,6,8,4,5,5,8,6,8,8,4,6,8,10,7,6,7,8,9,6,7,7,12,5,9,8,6,7,12,5,9,
%U A262985 6,9,6,11,9,11,5,6,10,8,7,9,11,5,7,7,8,7,9,8,8,9,6,7,9,7,10,9,4,6,6,7,9
%N A262985 Number of ordered ways to write n as 2^x + phi(y^2) + z*(z+1)/2 with x, y and z positive integers, where phi(.) is Euler's totient function given by A000010.
%C A262985 Conjecture: a(n) > 0 for all n > 3.
%C A262985 We have verified this for n up to 1.3*10^8.
%H A262985 Zhi-Wei Sun, <a href="/A262985/b262985.txt">Table of n, a(n) for n = 1..10000</a>
%e A262985 a(4) = 1 since 4 = 2 + phi(1^2) + 1*2/2.
%e A262985 a(5) = 1 since 5 = 2 + phi(2^2) + 1*2/2.
%e A262985 a(8) = 1 since 8 = 2^2 + phi(1^2) + 2*3/2.
%e A262985 a(36) = 2 since 36 = 2 + phi(3^2) + 7*8/2 = 2^5 + phi(1^2) + 2*3/2.
%t A262985  f[n_]:=EulerPhi[n^2]
%t A262985 TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
%t A262985 Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[TQ[n-f[x]-2^y],r=r+1], {y,1,Log[2,n-f[x]]}]; Label[aa];Continue,{x,1,n}];Print[n," ",r];Continue,{n,1,100}]
%Y A262985 Cf. A000010, A000079, A000217, A000290, A002618, A262976, A262980, A262982.
%K A262985 nonn
%O A262985 1,6
%A A262985 _Zhi-Wei Sun_, Oct 06 2015