

A211190


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


1



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
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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 nabc 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

ZhiWei Sun, Table of n, a(n) for n = 1..5000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205210 [MR96i:11106].
ZhiWei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
ZhiWei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 20122017.


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<3Mod[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[n1]==True&&pr[n+1]==True
a[n_]:=a[n]=Sum[If[p[j]==True&&p[k]==True&&q[2n2Prime[j]3Prime[k]]==True, 1, 0], {j, 1, PrimePi[n]}, {k, 1, PrimePi[(2n2Prime[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

ZhiWei Sun, Feb 03 2013


STATUS

approved



