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

2,1

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

Links

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

Formula

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

Example

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.

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 *)

Prog

(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<a, m=a)); print1(m, ", "))
(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

Crossrefs

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

Sequence in context: A239476 A117367 A080184 * A092386 A117369 A117366

Adjacent sequences: A052245 A052246 A052247 * A052249 A052250 A052251

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved