A085393 -id:A085393 - cp's OEIS frontend

A014673 Smallest prime factor of greatest proper divisor of n.

1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 2, 1, 3, 1, 2, 7, 11, 1, 2, 5, 13, 3, 2, 1, 3, 1, 2, 11, 17, 7, 2, 1, 19, 13, 2, 1, 3, 1, 2, 3, 23, 1, 2, 7, 5, 17, 2, 1, 3, 11, 2, 19, 29, 1, 2, 1, 31, 3, 2, 13, 3, 1, 2, 23, 5, 1, 2, 1, 37, 5, 2, 11, 3, 1, 2, 3, 41, 1, 2, 17, 43, 29, 2, 1, 3, 13, 2, 31, 47

Offset: 1

Reinhard Zumkeller, Jun 24 2003

nonn

For n > 1: a(n) = 1 iff n is prime; a(A001358(n)) = A084127(n); a(A025475(n)) = A020639(A025475(n)). [corrected by Peter Munn, Feb 19 2017]
When n is composite, this is the 2nd factor when n is written as a product of primes in nondecreasing order. For example, 12 = 2*2*3, so a(12) = 2. - Peter Munn, Feb 19 2017
For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(2) = 7. - Peter Munn, Jul 12 2019

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A001358, A020639, A025475, A027746, A032742, A054576, A084127, A085392, A085393, A115561, A117357, A117358, A281889.

PrimeFactors[ n_ ] := Flatten[ Table[ # [ [ 1 ] ], {1} ] & /@ FactorInteger[ n ] ]; f[ n_ ] := Block[ {gpd = Divisors[ n ][ [ -2 ] ]}, If[ gpd == 1, 1, PrimeFactors[ gpd ][ [ 1 ] ] ] ]; Table[ If[ n == 1, 1, f[ n ] ], {n, 1, 95} ]
(* Second program: *)
Table[If[Or[PrimeQ@ n, n == 1], 1, FactorInteger[n/SelectFirst[Prime@ Range@ PrimePi[Sqrt@ n], Divisible[n, #] &]][[1, 1]] ], {n, 94}] (* Michael De Vlieger, Aug 14 2017 *)

lpf(n)=if(n>1,factor(n)[1,1],1)
a(n)=lpf(n/lpf(n)) \\ Charles R Greathouse IV, May 09 2013

a(n)=if(n<4||isprime(n),return(1)); my(f=factor(n)); if(f[1,2]>1, f[1,1], f[2,1]) \\ Charles R Greathouse IV, May 09 2013

(define (A014673 n) (A020639 (/ n (A020639 n)))) ;; Code for A020639 given under that entry - Antti Karttunen, Aug 12 2017

a(n) = A020639(A032742(n)).
A117357(n) = A020639(A054576(n)); A117358(n) = A032742(A054576(n)) = A054576(n)/A117357(n). - Reinhard Zumkeller, Mar 10 2006
If A001222(n) >= 2, a(n) = A027746(n,2), otherwise a(n) = 1. - Peter Munn, Jul 13 2019

A085392 a(n) = largest prime divisor of n, or 1 if n is 1 or a prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 3, 1, 7, 5, 2, 1, 3, 1, 5, 7, 11, 1, 3, 5, 13, 3, 7, 1, 5, 1, 2, 11, 17, 7, 3, 1, 19, 13, 5, 1, 7, 1, 11, 5, 23, 1, 3, 7, 5, 17, 13, 1, 3, 11, 7, 19, 29, 1, 5, 1, 31, 7, 2, 13, 11, 1, 17, 23, 7, 1, 3, 1, 37, 5, 19, 11, 13, 1, 5, 3, 41, 1, 7, 17, 43, 29, 11, 1, 5, 13

Offset: 1

Robert G. Wilson v and Reinhard Zumkeller, Jun 26 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006530, A014673, A032742, A085393.

a085392 = a006530 . a032742  -- Reinhard Zumkeller, Oct 03 2012

A085392 := proc(n) max( op(numtheory[divisors](n) minus {n})) ; A006530(%) ;
end proc:
seq(A085392(n),n=1..50) ; # R. J. Mathar, Jun 26 2011

PrimeFactors[n_] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; f[n_] := Block[{gpd = Divisors[n][[ -2]]}, If[gpd == 1, 1, PrimeFactors[gpd][[ -1]] ]]; Table[ If[n == 1, 1, f[n]], {n, 1, 95}]
Join[{1},Table[FactorInteger[Divisors[n][[-2]]][[-1,1]],{n,2,120}]] (* Harvey P. Dale, Jul 02 2019 *)
a[n_] := If[CompositeQ[n], FactorInteger[n][[-1, 1]], 1]; Array[a, 100] (* Amiram Eldar, Jun 19 2022 *)

gpd(n) = if (n==1, 1, n/factor(n)[1,1]);
gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
a(n) = gpf(gpd(n)); \\ Michel Marcus, Apr 08 2018

a(n) = A006530(A032742(n)). - R. J. Mathar, Jun 26 2011

Definition corrected by N. J. A. Sloane, Jul 02 2019. Also deleted an incorrect comment.

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A014673 Smallest prime factor of greatest proper divisor of n.

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A085392 a(n) = largest prime divisor of n, or 1 if n is 1 or a prime.

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