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
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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