

A306382


Positive integers not representable as Pen(x) + Pen(y) + 2*Pen(z), where x, y, z are nonnegative integers, and Pen(k) denotes the pentagonal number k*(3k1)/2.


3



9, 18, 21, 28, 31, 39, 43, 55, 69, 74, 89, 90, 98, 109, 111, 113, 134, 135, 144, 193, 202, 214, 230, 243, 260, 265, 273, 275, 310, 510, 553, 698, 699, 749, 773, 780, 865, 878, 945, 965, 1219, 1398, 1413, 2153, 2168, 2335, 2828, 3178, 3793
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OFFSET

1,1


COMMENTS

Conjecture 1: This sequence only has 49 terms as listed.
Conjecture 2: Any integer n > 6471 can be written as Pen(x) + Pen(y) + 3*Pen(z) with x,y,z nonnegative integers. Any integer n > 7727 can be written as Pen(x) + 2*Pen(y) + 2*Pen(z) with x,y,z nonnegative integers. Also, any integer n > 4451 can be written as Pen(x) + 2*Pen(y) + 4*Pen(z) with x,y,z nonnegative integers.
Conjecture 3: Let N(1) = 5928, N(4) = 2761, N(8) = 8868 and N(11) = 9929. For each c among 1, 4, 8, 11, any integer n > N(c) can be written as x*(3x+1)/2 + y*(3y+1) + c*z*(3z+1)/2 with x,y,z nonnegative integers.
Conjecture 4: Any integer n > 5544 can be written as x*(3x+1)/2 + y*(3y+1)/2 + 3z*(3z+1)/2 with x,y,z nonnegative integers. Any integer n > 7093 can be written as x*(3x+1)/2 + 3*y*(3y+1)/2 + 2z*(3z+1) with x,y,z nonnegative integers. Also, any integer n > 8181 can be written as x*(3x+1)/2 + 2y*(3y+1) + 3z*(3z+1) with x,y,z nonnegative integers.
Conjecture 5: For each positive integer m, there are only finitely many positive integers not representable as x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 with x,y,z in the set {m, m+1, ...}.
See also A306383 for similar conjectures.


LINKS

Georg Fischer, Table of n, a(n) for n = 1..49
ZhiWei Sun, Universal sums of three quadratic polynomials, Sci. China Math., in press.


EXAMPLE

a(1) = 9 since the set {Pen(x) + Pen(y) + 2*Pen(z): x,y,z = 0,1,2,...} contains 1..8 but it does not contain 9.


MATHEMATICA

PenQ[n_]:=PenQ[n]=IntegerQ[Sqrt[24n+1]]&&(n==0Mod[Sqrt[24n+1]+1, 6]==0);
tab={}; Do[Do[If[PenQ[mx(3x1)y(3y1)/2], Goto[aa]], {x, 0, (Sqrt[12m+1]+1)/6}, {y, 0, (Sqrt[12(mx(3x1))+1]+1)/6}]; tab=Append[tab, m]; Label[aa], {m, 1, 5000}]; Print[tab]


CROSSREFS

Cf. A000217, A000326, A005449, A008443, A306383.
Sequence in context: A015785 A316438 A208135 * A222623 A141469 A046412
Adjacent sequences: A306379 A306380 A306381 * A306383 A306384 A306385


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 11 2019


STATUS

approved



