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A208246 Number of ways to write n = p+q with p prime or practical, and q-4, 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 (list; graph; refs; listen; history; text; internal format)
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, 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
Sequence in context: A336313 A095139 A109038 * A320857 A211664 A182434
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 11 2013
STATUS
approved

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)