|
%I #32 Dec 17 2017 10:06:18
%S 3,5,7,17,19,29,31,41,79,89,127,197,199,271,307,379,449,461,463,521,
%T 701,727,811,859,881,919,929,967,991,1217,1231,1289,1301,1409,1471,
%U 1481,1483,1567,1721,1889,1951,1999,2129,2393,2441,2549,2551,2729,2753,2861,2969,3041,3079,3319,3329,3331,3499,3739,3761,4049
%N Primes p with p-1 and p+1 both practical: "Sandwich of the first kind"
%C 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.
%C 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.
%C 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.
%C This conjecture is similar to Firoozbakht's conjecture for primes.
%H Zhi-Wei Sun, <a href="/A210479/b210479.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;20e70044.1301">Sandwiches with primes and practical numbers</a>, a message to Number Theory List, Jan. 13, 2013.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(1)=3 since 2 and 4 are practical.
%e a(2)=5 since 4 and 6 are practical.
%t f[n_]:=f[n]=FactorInteger[n]
%t Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t 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}]
%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t n=0
%t Do[If[pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True,n=n+1;Print[n," ",Prime[k]]],{k,1,100}]
%o (PARI) is_A210479(p)={is_A005153(p-1) && is_A005153(p+1) && isprime(p)} \\ _M. F. Hasler_, Jan 23 2013
%o (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
%Y Cf. A005153, A208249, A209236, A210480, A258838.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jan 23 2013
|
