cp's OEIS Frontend

A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.

%I A257497 #8 Apr 27 2015 04:04:27
%S A257497 1,2,2,2,3,2,3,3,3,2,2,2,5,2,3,4,4,4,2,2,3,4,6,3,2,5,7,5,2,4,3,5,4,3,
%T A257497 4,4,6,5,3,3,5,4,5,2,2,5,4,4,2,3,5,5,6,1,4,5,4,3,3,7,4,2,5,2,5,4,2,4,
%U A257497 3,6,4,5,9,4,3,3,4,8,2,4,5,3,5,1,5,4,1,5,3,2,4,6,6,3,5,4,6,5,5,5
%N A257497 Number of ordered ways to write n as the sum of a term of A257121 and a positive generalized pentagonal number.
%C A257497 Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 54, 84, 87, 109, 174, 252, 344, 1234, 1439, 2924.
%C A257497 This implies the Twin Prime Conjecture.
%H A257497 Zhi-Wei Sun, <a href="/A257497/b257497.txt">Table of n, a(n) for n = 1..20000</a>
%H A257497 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1504.01608">Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c)</a>, arXiv:1504.01608 [math.NT], 2015.
%e A257497 a(1439) = 1 since 1439 = 1424 + 15 = floor(4274/3) + (-3)*(3*(-3)-1)/2 with {3*4274-1,3*4274+1} = {12821,12823} a twin prime pair.
%e A257497 a(2924) = 1 since 2924 = 2334 + 590 = floor(7004/3) + 20*(3*20-1)/2 with {3*7004-1, 3*7004+1} = {21011,21013} a twin prime pair.
%t A257497 TQ[n_]:=PrimeQ[3n-1]&&PrimeQ[3n+1]
%t A257497 PQ[n_]:=TQ[3*n]||TQ[3*n+1]||TQ[3n+2]
%t A257497 SQ[n_]:=IntegerQ[Sqrt[24n+1]]
%t A257497 Do[m=0;Do[If[PQ[x]&&SQ[n-x],m=m+1],{x,0,n-1}];
%t A257497 Print[n," ",m];Continue,{n,1,100}]
%Y A257497 Cf. A001318, A014574, A257121, A256071, A256707, A257317, A257474.
%K A257497 nonn
%O A257497 1,2
%A A257497 _Zhi-Wei Sun_, Apr 26 2015