A258836 Least practical number q with q-1 and q+1 twin prime such that n = q'/q for some practical number q' with q'-1 and q'+1 twin prime.

4, 6, 4, 18, 6, 12, 6, 30, 12, 6, 18, 6, 150, 30, 4, 12, 60, 4, 12, 12, 42, 30, 240, 18, 6, 12, 4, 270, 12, 6, 42, 6, 6, 30, 12, 12, 180, 6, 60, 6, 30, 150, 30, 30, 4, 18, 2550, 4, 18, 12, 42, 6, 150, 30, 12, 60, 4, 6, 60, 4, 462, 180, 1230, 18, 30, 108, 60, 180, 12, 6, 30, 6, 570, 420, 462, 180, 6, 4, 198, 42, 522, 600, 1050, 42, 12, 12, 4, 60, 432, 18, 12, 60, 30, 60, 6, 12, 150, 60, 30, 6

Offset: 1

Author

Zhi-Wei Sun, Jun 11 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. Moreover, any positive rational number r can be written as q'/q, where q and q' are terms of A258838 (i.e., q is practical with q-1 and q+1 twin prime, and q' is practical with q'-1 and q'+1 twin prime).

This implies that there are infinitely many "sandwiches of the second kind" (i.e., triples {q-1,q,q+1} with q practical and q-1 and q+1 twin prime).

I have verified the conjecture for all those rational numbers r = n/m with m,n = 1,...,1000. -Zhi-Wei Sun, Jun 15 2015

Examples

			a(1) = 4 since 1 = 4/4 with 4 practical and 4-1 and 4+1 twin prime.
a(2) = 6 since 2 = 12/6, 6 is practical with 6-1 and 6+1 twin prime, and 12 is practical with 12-1 and 12+1 twin prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Checking the conjecture for r = n/m with 1 <= n <= m <= 1000
• 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, A001359, A006512, A005153, A210479, A258803, A258811, A258838.

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)
SW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&pr[n]
Do[k=0;Label[bb];k=k+1;If[PrimeQ[Prime[k]+2]&&pr[Prime[k]+1]&&SW[n*(Prime[k]+1)],Goto[aa],Goto[bb]];
Label[aa];Print[n," ",Prime[k]+1];Continue,{n,1,100}]