A257922 Practical numbers m with m-1 and m+1 both prime, and prime(m)-1 and prime(m)+1 both practical.

4, 522, 1932, 5100, 6132, 6552, 8220, 18312, 18540, 22110, 29568, 45342, 70488, 70950, 92220, 105360, 109662, 114600, 116532, 117192, 123552, 128982, 131838, 132762, 136710, 148302, 149160, 166848, 177012, 183438, 197340, 206280, 233550, 235008, 257868, 272808, 273900, 276780, 279708, 286590

Offset: 1

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

• nonn

Comments

Conjecture: The sequence contains infinitely many terms. In other words, there are infinitely many positive integers n such that {prime(n)-1, prime(n), prime(n)+1} is a "sandwich of the first kind" (A210479) and {n-1, n, n+1} is a "sandwich of the second kind" (A258838).

This implies that there are infinitely many sandwiches of the first kind and also there are infinitely many sandwiches of the second kind.

Examples

			a(1) = 4 since 4 is paractical with 4-1 and 4+1 twin prime, and prime(4)-1 = 6 and prime(4)+1 = 8 are both practical.
a(2) = 522 since 522 is paractical with 522-1 and 522+1 twin prime, and prime(522)-1 = 3738 and prime(522)+1 = 3740 are both practical.
		

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 involving primes and quadratic forms, arXiv:1211.1588.

Crossrefs

Cf. A000040, A005153, A014574, A209236, A210479, A257924, A258836, A258838, A259539.

Programs

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)
n=0;Do[If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&pr[Prime[Prime[k]+1]-1]&&pr[Prime[Prime[k]+1]+1],n=n+1;Print[n," ",Prime[k]+1]],{k,1,24962}]