OFFSET
1,1
COMMENTS
When p is a prime with p-1 and p+1 both practical, {p-1, p, p+1} is a sandwich of the first kind introduced by Zhi-Wei Sun. He conjectured that there are infinitely many such sandwiches. See also A210480 for a strong conjecture involving terms in the current sequence.
No term can be congruent to 1 or -1 modulo 12. In fact, if p>3 and 12|p-1, then neither 3 nor 4 divides p+1, hence p+1 is not practical since 4 is not a sum of some distinct divisors of p+1. Similarly, if 12|p+1 then p-1 is not practical.
Conjecture: The sequence a(n)^(1/n) (n=9,10,...) is strictly decreasing to the limit 1. Also, if {b(n)-1,b(n),b(n)+1} is the n-th sandwich of the second kind, then the sequence b(n)^(1/n) (n=1,2,3,...) is strictly decreasing to the limit 1.
This conjecture is similar to Firoozbakht's conjecture for primes.
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)=3 since 2 and 4 are practical.
a(2)=5 since 4 and 6 are 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)
n=0
Do[If[pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True, n=n+1; Print[n, " ", Prime[k]]], {k, 1, 100}]
PROG
(PARI) is_A210479(p)={is_A005153(p-1) && is_A005153(p+1) && isprime(p)} \\ M. F. Hasler, Jan 23 2013
(PARI) A210479(n, print_all=0)={forprime(p=3, , is_A005153(p-1) & is_A005153(p+1) & !(print_all & print1(p", ")) & !n-- & return(p))} \\ M. F. Hasler, Jan 23 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 23 2013
STATUS
approved