A208244 Number of ways to write n as the sum of a practical number (A005153) and a triangular number (A000217).
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, 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, 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
Offset: 1
Examples
a(15)=1 since 15=12+3 with 12 a practical number and 3 a triangular number.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
- Giuseppe Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
- Sai Teja Somu and Duc Van Khanh Tran, On Sums of Practical Numbers and Polygonal Numbers, arXiv:2403.13533 [math.NT], 2024.
- Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), 65-76.
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017.
Programs
-
Mathematica
f[n_]:=f[n]=FactorInteger[n] Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2]) 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}] pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0) a[n_]:=a[n]=Sum[If[pr[n-k(k+1)/2]==True,1,0],{k,0,(Sqrt[8n+1]-1)/2}] Do[Print[n," ",a[n]],{n,1,100}]
Comments