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A208249
Number of primes p with n < p < 2n, such that p-1 and p+1 are both practical.
17
0, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 2, 2, 2, 3, 4, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2
OFFSET
1,4
COMMENTS
Conjecture: a(n)>0 for all n>8.
Zhi-Wei Sun also made the following conjectures:
(1) For each integer n>6 there is a practical number q with n<q<2n such that q-1 and q+1 are both prime.
(2) For any integer n>231 there is a prime p with n<p<2n-1 such that p+2 is prime, and p-1 and p+1 are all practical.
(3) There are infinitely many twin prime pairs {p,p+2} with p-1,p+1,p+3 all practical.
(4) Any odd number n>1 can be written as p+q (p,q>0) with p practical and p^2+q^2 prime.
LINKS
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 2012-2017.
FORMULA
a(n) = card { p in A000040 | n < p < 2n, p-1 and p+1 are both practical }.
EXAMPLE
a(9)=1 since 17 is the only prime 9<p<18 with p-1 and p+1 both 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[PrimeQ[n+k] == True && pr[n+k-1] == True && pr[n+k+1] == True, 1, 0], {k, 1, n-1}]; Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 12 2013
STATUS
approved