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%I #28 Apr 24 2024 12:29:24
%S 1,2,1,2,2,1,3,2,2,1,2,3,1,2,1,3,2,3,3,1,3,3,3,2,2,2,3,2,3,4,3,2,4,3,
%T 2,3,3,3,3,4,2,4,3,2,3,4,2,4,3,1,4,3,2,3,2,4,6,2,2,4,4,1,5,4,2,4,4,3,
%U 4,4,2,4,3,2,5,3,2,4,4,2,5,4,2,6,4,3,5,3,1,6,3,3,5,5,3,5,3,3,5,4
%N Number of ways to write n as the sum of a practical number (A005153) and a triangular number (A000217).
%C Conjecture: a(n)>0 for all n>0.
%C The author has verified this for n up to 10^8, and also guessed the following refinement: If n>6 is not among 20, 104, 272, 464, 1664, then n can be written as p+q with p an even practical number and q a positive triangular number.
%C Somu and Tran (2024) proved the conjecture that a(n)>0 for n>0. - _Duc Van Khanh Tran_, Apr 24 2024
%H Zhi-Wei Sun, <a href="/A208244/b208244.txt">Table of n, a(n) for n = 1..50000</a>
%H Giuseppe Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].
%H Sai Teja Somu and Duc Van Khanh Tran, <a href="https://arxiv.org/abs/2403.13533">On Sums of Practical Numbers and Polygonal Numbers</a>, arXiv:2403.13533 [math.NT], 2024.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/109p.pdf">On sums of primes and triangular numbers</a>, J. Comb. Number Theory 1(2009), 65-76.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arxiv:1211.1588 [math.NT], 2012-2017.
%e a(15)=1 since 15=12+3 with 12 a practical number and 3 a triangular number.
%t f[n_]:=f[n]=FactorInteger[n]
%t Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
%t Con[n_]:=Con[n]=Sum[If[Part[Part[f[n],s+1],1]<=DivisorSigma[1,Product[Pow[n,i],{i,1,s}]]+1,0,1],{s,1,Length[f[n]]-1}]
%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
%t a[n_]:=a[n]=Sum[If[pr[n-k(k+1)/2]==True,1,0],{k,0,(Sqrt[8n+1]-1)/2}]
%t Do[Print[n," ",a[n]],{n,1,100}]
%Y Cf. A000040, A005153, A208243, A132399, A187785.
%K nonn,look
%O 1,2
%A _Zhi-Wei Sun_, Jan 11 2013
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