A208249 Number of primes p with n < p < 2n, such that p-1 and p+1 are both practical.

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

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

Cf. A000040, A001097, A005153, A208243, A208244, A208246.

Sequence in context: A191336 A277349 A078807 * A329985 A029422 A351356

Adjacent sequences: A208246 A208247 A208248 * A208250 A208251 A208252

Keyword

nonn

Author

Zhi-Wei Sun, Jan 12 2013

Status

approved