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 A208244 Number of ways to write n as the sum of a practical number (A005153) and a triangular number (A000217). 20
 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 (list; graph; refs; listen; history; text; internal format)
 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. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..50000 G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106]. Z. W. 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

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Last modified October 13 20:38 EDT 2019. Contains 327981 sequences. (Running on oeis4.)