cp's OEIS Frontend

A306225 Number of ways to write n as w + x^5 + pen(y) + pen(z), where w is 0 or 1, and x,y,z are integers with x >= w and pen(y) < pen(z), and where pen(m) denotes the pentagonal number m*(3*m-1)/2.

%I A306225 #12 Jan 30 2019 09:08:45
%S A306225 1,2,3,2,2,2,4,4,4,2,1,2,3,4,3,3,4,3,3,2,2,3,3,4,2,2,4,5,5,2,2,1,3,4,
%T A306225 5,4,5,6,6,6,6,7,4,4,4,3,5,5,6,4,3,6,5,5,5,4,5,6,9,7,4,4,5,5,3,5,4,4,
%U A306225 4,5,4,6,8,7,5,2,6,5,8,6,3,3,5,7,6,4,3,3,4,5,5,6,7,9,5,4,4,5,6,3
%N A306225 Number of ways to write n as w + x^5 + pen(y) + pen(z), where w is 0 or 1, and x,y,z are integers with x >= w and pen(y) < pen(z), and where pen(m) denotes the pentagonal number m*(3*m-1)/2.
%C A306225 Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer can be written as the sum of two fifth powers of nonnegative integers one of which is 0 or 1, and two distinct generalized pentagonal numbers.
%C A306225 We have verified a(n) > 0 for all n = 1..2*10^6. The conjecture implies that the set A = {x^5 + pen(y): x = 0,1,2,... and y is an integer} is an additive basis of order two (i.e., the sumset A + A coincides with {0,1,2,...}).
%C A306225 See also A306227 for a similar conjecture.
%H A306225 Zhi-Wei Sun, <a href="/A306225/b306225.txt">Table of n, a(n) for n = 1..10000</a>
%e A306225 a(11) = 1 with 11 = 1 + 1^5 + pen(-1) + pen(-2).
%e A306225 a(1000) = 1 with 1000 = 0 + 2^5 + pen(8) + pen(-24).
%e A306225 a(5104) = 1 with 5104 = 1 + 3^5 + pen(-3) + pen(57).
%e A306225 a(8196) = 1 with 8196 = 0 + 2^5 + pen(48) + pen(-56).
%e A306225 a(9537) = 1 with 9537 = 1 + 6^5 + pen(17) + pen(30).
%e A306225 a(15049) = 1 with 15049 = 0 + 6^5 + pen(-44) + pen(54).
%e A306225 a(16775) = 1 with 16775 = 1 + 5^5 + pen(-17) + pen(94).
%t A306225 PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
%t A306225 tab={};Do[r=0;Do[If[PenQ[n-x-y^5-z(3z-1)/2],r=r+1],{x,0,Min[1,(n-1)/2]},{y,x,(n-1-x)^(1/5)},{z,-Floor[(Sqrt[12(n-1-x-y^5)+1]-1)/6],(Sqrt[12(n-1-x-y^5)+1]+1)/6}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A306225 Cf. A000584, A001318, A270920, A306227.
%K A306225 nonn
%O A306225 1,2
%A A306225 _Zhi-Wei Sun_, Jan 30 2019