A208246 Number of ways to write n = p+q with p prime or practical, and q-4, q, q+4 all practical

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

Offset

1,13

Comments

Conjecture: a(n)>0 for all n>8.

Zhi-Wei Sun also made some similar conjectures, below are few examples.

(1) Each integer n>3 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 q-8, q, q+8 all practical.

(3) The interval [n,2n) contains a practical number p with p-n 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

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Zhi-Wei Sun, Conjectures on representations involving primes, arxiv:1211.1588 [math.NT], 2012-2017.

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<3||Mod[n, 2]+Con[n]==0)
a[n_]:=a[n]=Sum[If[pr[k]==True&&pr[k-4]==True&&pr[k+4]==True&&(PrimeQ[n-k]==True||pr[n-k]==True), 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A005153, A208243, A208244, A219842, A220272.

Sequence in context: A336313 A095139 A109038 * A320857 A211664 A182434

Adjacent sequences: A208243 A208244 A208245 * A208247 A208248 A208249

Keyword

nonn

Author

Zhi-Wei Sun, Jan 11 2013

Status

approved