A052248 Greatest prime divisor of all composite numbers between p and next prime.

2, 3, 5, 3, 7, 3, 11, 13, 5, 17, 19, 7, 23, 17, 29, 5, 31, 23, 3, 37, 41, 43, 47, 11, 17, 53, 3, 37, 61, 43, 67, 23, 73, 5, 31, 79, 83, 43, 89, 5, 61, 3, 97, 11, 103, 109, 113, 19, 29, 79, 5, 83, 127, 131, 89, 5, 137, 139, 47, 97, 151, 103, 13, 157, 163, 167, 173, 29, 13

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Or, largest of all prime factors of the numbers between prime(n) and prime(n+1).

a(n) = 3, 5, 7, 11, 13 iff prime(n) is in A059960, A080185, A080186, A080187, A080188 respectively. This sequence defines a mapping f of primes > 2 to primes (cf. A080189) and f(p) < p holds for all p > 2. - Klaus Brockhaus, Feb 10 2003

a(n) = A006530(A061214(n)). - Reinhard Zumkeller, Jun 22 2011

Examples

			a(8) = 11 since 20 = 2*2*5, 21 = 3*7, 22 = 2*11 are the numbers between prime(8) = 19 and prime(9) = 23.
For n=9, n-th prime is 23, composites between 23 and next prime are 24 25 26 27 29 of which largest prime divisor is 13, so a(9)=13.
		

Links

• T. D. Noe, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A006530, A059960, A080185, A080186, A080187, A080188, A080189, A052180, A065091.

Programs

Haskell

a052248 n = a052248_list !! (n-2)
a052248_list = f a065091_list where
   f (p:ps'@(p':ps)) = (maximum $ map a006530 [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011

Mathematica

g[n_] := Block[{t = Range[Prime[n] + 1, Prime[n + 1] - 1]}, Max[First /@ Flatten[ FactorInteger@t, 1]]]; Table[ g[n], {n, 2, 72}] (* Robert G. Wilson v, Feb 08 2006 *)
cmp[{a_,b_}]:=Max[Flatten[FactorInteger/@Range[a+1,b-1],1][[All,1]]]; cmp/@ Partition[ Prime[Range[2,80]],2,1] (* Harvey P. Dale, May 16 2020 *)

PARI

forprime(p=3,360,q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m
				

Formula

a(n) = max(prime(n) < k < prime(n+1), A006530(k)).