

A208246


Number of ways to write n = p+q with p prime or practical, and q4, q, q+4 all practical


19



0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 4, 4, 3, 4, 5, 4, 3, 5, 6, 4, 3, 5, 7, 5, 4, 6, 8, 4, 3, 5, 8, 4, 2, 4, 8, 5, 3, 4, 7, 4, 3, 5, 7, 3, 2, 4, 6, 5, 4, 4, 7, 5, 4, 5, 7, 4, 2, 4, 7, 5, 3, 4, 6, 4, 4, 6, 6, 3, 2, 5, 6, 4, 4, 5, 7, 5, 5, 7, 8, 2, 2, 6, 8, 5, 3, 4, 7
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OFFSET

1,13


COMMENTS

Conjecture: a(n)>0 for all n>8.
ZhiWei Sun also made some similar conjectures, below are few examples.
(1) Each integer n>2 can be written as p+q with p prime or practical, and q and q+2 both practical.
(2) Any integer n>12 can be written as p+q with p prime or practical, and q8, q, q+8 all practical.
(3) The interval [n,2n) contains a practical number p with pn a triangular number.
(4) Any integer n>1 can be written as x^2+y (x,y>0) with 2x and 2xy both practical.
Note that if x>=y>0 with x practical then xy is also practical.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..20000
ZhiWei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588.


EXAMPLE

a(11)=1 since 11=3+8 with 3 prime, and 4, 8, 12 all practical.
a(12)=1 since 12=4+8 with 4, 8, 12 all practical.


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)
a[n_]:=a[n]=Sum[If[pr[k]==True&&pr[k4]==True&&pr[k+4]==True&&(PrimeQ[nk]==Truepr[nk]==True), 1, 0], {k, 1, n1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]


CROSSREFS

Cf. A005153, A208243, A208244, A219842, 220272.
Sequence in context: A273165 A095139 A109038 * A320857 A211664 A182434
Adjacent sequences: A208243 A208244 A208245 * A208247 A208248 A208249


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 11 2013


STATUS

approved



