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

3, 5, 7, 17, 19, 29, 31, 41, 79, 89, 127, 197, 199, 271, 307, 379, 449, 461, 463, 521, 701, 727, 811, 859, 881, 919, 929, 967, 991, 1217, 1231, 1289, 1301, 1409, 1471, 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

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

Cf. A005153, A208249, A209236, A210480, A258838.

Sequence in context: A258195 A110358 A038971 * A045400 A163425 A191038

Adjacent sequences: A210476 A210477 A210478 * A210480 A210481 A210482

Keyword

nonn

Author

Zhi-Wei Sun, Jan 23 2013

Status

approved