OFFSET
1,1
COMMENTS
The author introduced two kinds of "sandwiches" in 2013. The conjecture in A258836 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the sequence contains infinitely many terms.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)
EXAMPLE
a(1) = 4 since 4 is practical with 4-1 and 4+1 twin prime.
a(2) = 6 since 6 is practical with 6-1 and 6+1 twin prime.
a(3) = 12 since 12 is practical with 12-1 and 12+1 twin prime.
MATHEMATICA
f[n_]:=FactorInteger[n]
Pow[n_, i_]:=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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_]:=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n]
n=0; Do[If[SW[m], n=n+1; Print[n, " ", m]], {m, 1, 2730}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 12 2015
STATUS
approved