cp's OEIS Frontend

A343460 Number of ways to write n as x^6 + y^3 + z*(3*z+1)/2 + 2^k, where x and y are nonnegative integers, z is an integer and k is a positive integer.

%I A343460 #14 Apr 18 2021 06:12:40
%S A343460 0,1,3,5,6,5,4,4,6,9,8,6,5,5,6,7,11,11,7,5,5,5,5,8,8,5,4,5,7,7,10,11,
%T A343460 7,8,8,8,8,9,10,8,6,7,10,10,10,7,6,7,4,5,7,6,5,4,7,8,6,5,7,8,7,6,3,5,
%U A343460 8,12,15,13,12,10,9,11,17,18,13,9,6,9,11,16
%N A343460 Number of ways to write n as x^6 + y^3 + z*(3*z+1)/2 + 2^k, where x and y are nonnegative integers, z is an integer and k is a positive integer.
%C A343460 Conjecture: a(n) > 0 for all n > 1.
%C A343460 We have verified a(n) > 0 for n = 2..10^7.
%C A343460 Conjecture verified up to 10^10. - _Giovanni Resta_, Apr 18 2021
%H A343460 Zhi-Wei Sun, <a href="/A343460/b343460.txt">Table of n, a(n) for n = 1..10000</a>
%e A343460 a(2) = 1 with 2 = 0^6 + 0^3 + 0*(3*0+1)/2 + 2^1.
%e A343460 a(175) = 2 with 175 = 1^6 + 3^3 + (-10)*(3*(-10)+1)/2 + 2^1 = 2^6 + 4^3 + 3*(3*3+1)/2 + 2^5.
%e A343460 a(14553) = 1 with 14553 = 2^6 + 17^3 + (-80)*(3*(-80)+1)/2 + 2^4.
%t A343460 PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
%t A343460 tab={};Do[r=0;Do[If[PenQ[n-x^6-y^3-2^k],r=r+1],{x,0,(n-1)^(1/6)},{y,0,(n-x^6-1)^(1/3)},{k,1,Log[2,n-x^6-y^3]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A343460 Cf. A000079, A000578, A001014, A001318, A343397, A343411.
%K A343460 nonn
%O A343460 1,3
%A A343460 _Zhi-Wei Sun_, Apr 16 2021