cp's OEIS Frontend

A210479 Primes p with p-1 and p+1 both practical: "Sandwich of the first kind".

%I A210479 #34 Jul 22 2025 22:30:33
%S A210479 3,5,7,17,19,29,31,41,79,89,127,197,199,271,307,379,449,461,463,521,
%T A210479 701,727,811,859,881,919,929,967,991,1217,1231,1289,1301,1409,1471,
%U A210479 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 A210479 Primes p with p-1 and p+1 both practical: "Sandwich of the first kind".
%C A210479 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 A210479 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 A210479 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 A210479 This conjecture is similar to Firoozbakht's conjecture for primes.
%H A210479 Zhi-Wei Sun, <a href="/A210479/b210479.txt">Table of n, a(n) for n = 1..10000</a>
%H A210479 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 A210479 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 A210479 a(1)=3 since 2 and 4 are practical.
%e A210479 a(2)=5 since 4 and 6 are practical.
%t A210479 f[n_]:=f[n]=FactorInteger[n]
%t A210479 Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
%t A210479 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 A210479 pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
%t A210479 n=0
%t A210479 Do[If[pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True,n=n+1;Print[n," ",Prime[k]]],{k,1,100}]
%o A210479 (PARI) is_A210479(p)={is_A005153(p-1) && is_A005153(p+1) && isprime(p)} \\ _M. F. Hasler_, Jan 23 2013
%o A210479 (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 A210479 Cf. A005153, A208249, A209236, A210480, A258838.
%K A210479 nonn
%O A210479 1,1
%A A210479 _Zhi-Wei Sun_, Jan 23 2013