OFFSET
1,2
COMMENTS
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.
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,...}).
See also A306227 for a similar conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(11) = 1 with 11 = 1 + 1^5 + pen(-1) + pen(-2).
a(1000) = 1 with 1000 = 0 + 2^5 + pen(8) + pen(-24).
a(5104) = 1 with 5104 = 1 + 3^5 + pen(-3) + pen(57).
a(8196) = 1 with 8196 = 0 + 2^5 + pen(48) + pen(-56).
a(9537) = 1 with 9537 = 1 + 6^5 + pen(17) + pen(30).
a(15049) = 1 with 15049 = 0 + 6^5 + pen(-44) + pen(54).
a(16775) = 1 with 16775 = 1 + 5^5 + pen(-17) + pen(94).
MATHEMATICA
PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]];
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 30 2019
STATUS
approved