OFFSET
1,11
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.
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.
EXAMPLE
a(10)=1 since 2*10=5+2*3+3*3 with 3 and 5 terms of A210479.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 03 2013
STATUS
approved