A211190 Number of ways to write 2n = p+2q+3r with p,q,r terms of A210479.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 3, 4, 3, 3, 3, 4, 4, 5, 5, 5, 5, 4, 7, 6, 6, 7, 5, 6, 7, 7, 7, 7, 5, 5, 8, 6, 7, 8, 5, 8, 10, 9, 9, 11, 9, 8, 12, 9, 8, 10, 7, 7, 10, 8, 7, 9, 7, 6, 12, 8, 9, 11, 7, 8, 10, 8, 7, 11, 8, 7, 11, 7, 7, 10, 6, 5, 8, 7, 6, 10, 7, 7, 10, 7, 6, 11, 7, 7, 10, 5, 5, 10, 5

Offset: 1

Author

Zhi-Wei Sun, Feb 03 2013

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>8. Moreover, for positive integers a<=b<=c, all integers n>=3(a+b+c) with n-a-b-c even can be written as a*p+b*q+c*r with p,q,r terms of A210479, if and only if (a,b,c) is among the following 6 triples: (1,2,3), (1,2,4), (1,2,8), (1,2,9), (1,3,5), (1,3,8).

The author also conjectured that if n>8 is odd, different from 201 and 447, and not congruent to 1 or -1 modulo 12, then n can be written as a sum of three terms of A210479.

Examples

			a(10)=1 since 2*10=5+2*3+3*3 with 3 and 5 terms of A210479.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
• Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.

Crossrefs

Cf. A005153, A210479, A210480, A210681, A211165.

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)
p[k_]:=p[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True
q[n_]:=q[n]=PrimeQ[n]==True&&pr[n-1]==True&&pr[n+1]==True
a[n_]:=a[n]=Sum[If[p[j]==True&&p[k]==True&&q[2n-2Prime[j]-3Prime[k]]==True,1,0],{j,1,PrimePi[n]},{k,1,PrimePi[(2n-2Prime[j])/3]}]
Do[Print[n," ",a[n]],{n,1,100}]