A308621 - OEIS

%I #10 Jun 11 2019 10:44:50

%S 1,2,2,3,3,2,3,5,2,4,4,3,3,5,3,3,6,4,4,5,2,6,5,4,4,5,2,4,6,3,4,8,5,3,

%T 5,4,5,8,5,5,4,3,5,7,4,5,8,4,2,8,2,6,7,4,3,4,6,5,8,4,4,6,5,5,5,5,6,8,

%U 4,6,7,4,6,10,4,4,7,5,2,10,4,7,7,4,8,4,4,7,8,2,4,9,5,5,9,5,5,7,5,5

%N Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*5^(3d) with a,b,c,d nonnegative integers.

%C Conjecture: a(n) > 0 for all n > 0. Equivalently, any positive integer n can be written as a^2 + b*(b+1) + 2^c*5^(3d) with a,b,c,d nonnegative integers.

%C This was motivated by A308584, and we verified a(n) > 0 for all n = 1..2*10^8. Then, on the author's request, _Giovanni Resta_ verified the above conjecture for n up to 10^10. G. Resta also noted that 7272774228 cannot be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*250^d with a,b,c,d nonnegative integers.

%C See also A308566, A308584 and A308623 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A308621/b308621.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 1 with 1 = 0*1/2 + 0*1/2 + 2^0*5^(3*0).

%e a(78210) = 1 with 78210 = 85*86/2 + 385*386/2 + 2*5^3.

%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];

%t tab={};Do[r=0;Do[If[TQ[n-5^(3k)*2^m-x(x+1)/2],r=r+1],{k,0,Log[5,n]/3},{m,0,Log[2,n/5^(3k)]},{x,0,(Sqrt[4(n-5^(3k)*2^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

%Y Cf. A000079, A000217, A000351, A308566, A308584, A308623.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jun 11 2019