A267422 Largest prime factor of the largest squarefree practical number comprising n prime factors.

2, 3, 13, 167, 28219, 796481281, 634382430983400959, 402441068740409482305343048128921493, 161958813808922990180784918278423278413890512706478208244331277280870341

Offset: 1

Author

Frank M Jackson, Jan 14 2016

Keywords

• nonn

Comments

The prime factors of the largest squarefree practical number with n prime factors are the first n members of a(n). The largest squarefree practical number with 3 prime factors is 78 = 2*3*13 and the largest squarefree practical number with 5 prime factors is 367580694 = 2*3*13*167*28219, etc.

Because all primorial numbers (A002110) are practical, the prime factors of the smallest squarefree practical number with n prime factors are the first n members of the primes. Hence the smallest squarefree practical number with n prime factors is A002110(n). - Frank M Jackson, May 29 2023

Examples

			a(3) = 13 because there are only 4 squarefree practical numbers with 3 prime factors, namely 2*3*5 = 30, 2*3*7 = 42, 2*3*11 = 66 and 2*3*13 = 78. So 78 is the largest squarefree practical number with 3 prime factors and the largest prime factor is 13.
		

Links

• Frank M Jackson, Table of n, a(n) for n = 1..15
• Wikipedia, Practical number and Squarefree integer

Crossrefs

Cf. A002110, A005117, A005153, A265501.

Programs

Mathematica

lst={2}; Do[If[PrimeQ[f=DivisorSigma[1, Apply[Times, lst]]+1], AppendTo[lst, f], AppendTo[lst, NextPrime[f, -1]]], {8}]; lst
lst={2}; Do[AppendTo[lst, NextPrime[Times@@(#+1)&[lst]+2, -1]], {12}]; lst (* Frank M Jackson, May 29 2023 *)