cp's OEIS Frontend

A375249 Integers which cannot be partitioned into the sum of a hexagonal number plus a pentagonal number, nor a hexagonal number plus a square, nor a pentagonal number plus a square.

%I A375249 #23 Nov 09 2024 17:27:16
%S A375249 3,8,34,43,56,59,62,68,72,73,83,89,90,97,104,110,111,114,131,138,139,
%T A375249 148,152,163,164,167,168,186,193,200,203,205,207,222,227,228,229,233,
%U A375249 244,249,250,252,258,269,273,279,299,300,305,306,308,309,318,319,321,333,343,344,356,364,365
%N A375249 Integers which cannot be partitioned into the sum of a hexagonal number plus a pentagonal number, nor a hexagonal number plus a square, nor a pentagonal number plus a square.
%C A375249 Inspired by A160324 and analogous to A020757.
%C A375249 C. F. Gauss proved that all positive integers can be expressed as the sum of three triangular numbers. However, Zhi-Wei Sun (2009) has shown that there are 95 candidates for universal triples. This sequence looks at the {p4, p5, p6} triples and ask which integers require all three members to satisfy the sum.
%C A375249 Obviously, a(n) cannot be either a square, a pentagonal number, nor a hexagonal number.
%C A375249 There are more terms less than some integer in A020757 than in this sequence. In a sense, a square plus a pentagonal plus a hexagonal number is more efficient than the sum of three triangular numbers.
%C A375249 a(n) =~ 5.14 n^(.97).
%H A375249 Robert G. Wilson v, <a href="/A375249/b375249.txt">Table of n, a(n) for n = 1..10000</a>
%H A375249 Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:0905.0635 [math.NT], 2009-2015.
%e A375249 7 is not in the sequence since the third hexagonal number 6 plus the second square or pentagonal number sum to 7;
%e A375249 8 is in the sequence because s = {0, 1, 4}, p = {0, 1, 5}, and h = {0, 1, 6} with no two sets having members which sum to 8.
%t A375249 planeFiguratePi[r_,n_] := Floor[((r -4) +Sqrt[(r -4)^2 + 8n (r -2)])/(2 (r - 2))];
%t A375249 h = Table[PolygonalNumber[6, n], {n, 0, planeFiguratePi[6, 500]}];
%t A375249 p = Table[PolygonalNumber[5, n], {n, 0, planeFiguratePi[5, 500]}];
%t A375249 s = Table[PolygonalNumber[4, n], {n, 0, planeFiguratePi[4, 500]}];
%t A375249 Complement[ Range@ 500, Flatten[{Outer[Plus, h, p], Outer[Plus, h, s], Outer[Plus, p, s]} ]]
%Y A375249 Cf. A000290, A000326, A000384, A020757, A160324.
%K A375249 easy,nonn
%O A375249 1,1
%A A375249 _Robert G. Wilson v_, Aug 07 2024