

A208249


Number of primes p with n < p < 2n, such that p1 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
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OFFSET

1,4


COMMENTS

Conjecture: a(n)>0 for all n>8.
ZhiWei Sun also made the following conjectures:
(1) For each integer n>6 there is a practical number q with n<q<2n such that q1 and q+1 are both prime.
(2) For any integer n>231 there is a prime p with n<p<2n1 such that p+2 is prime, and p1 and p+1 are all practical.
(3) There are infinitely many twin prime pairs {p,p+2} with p1,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

ZhiWei Sun, Table of n, a(n) for n = 1..10000
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205210 [MR96i:11106].
ZhiWei Sun, Conjectures involving primes and quadratic forms, arxiv:1211.1588 [math.NT], 20122017.


FORMULA

a(n) = card { p in A000040  n < p < 2n, p1 and p+1 are both practical }.


EXAMPLE

a(9)=1 since 17 is the only prime 9<p<18 with p1 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<3Mod[n, 2]+Con[n]==0); a[n_] := a[n] = Sum[If[PrimeQ[n+k] == True && pr[n+k1] == True && pr[n+k+1] == True, 1, 0], {k, 1, n1}]; Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A001097, A005153, A208243, A208244, A208246.
Sequence in context: A191336 A277349 A078807 * A029422 A152800 A223730
Adjacent sequences: A208246 A208247 A208248 * A208250 A208251 A208252


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 12 2013


STATUS

approved



