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

1,2

Comments

Conjecture: a(n)>0 for all n>0.

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.

Somu and Tran (2024) proved the conjecture that a(n)>0 for n>0. - Duc Van Khanh Tran, Apr 24 2024

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.

Example

a(15)=1 since 15=12+3 with 12 a practical number and 3 a triangular number.

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}]

Crossrefs

Cf. A000040, A005153, A208243, A132399, A187785.

Sequence in context: A219607 A205451 A227899 * A272171 A160696 A152545

Adjacent sequences: A208241 A208242 A208243 * A208245 A208246 A208247

Keyword

nonn,look

Author

Zhi-Wei Sun, Jan 11 2013

Status

approved