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

1,11

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

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

Sequence in context: A086712 A125842 A029865 * A072644 A179864 A070082

Adjacent sequences: A211187 A211188 A211189 * A211191 A211192 A211193

Keyword

nonn

Author

Zhi-Wei Sun, Feb 03 2013

Status

approved