A092975 Consider all partitions of n into parts all of which are divisors of n; a(n) = maximal product of parts.

1, 2, 3, 4, 5, 9, 7, 16, 27, 32, 11, 81, 13, 128, 243, 256, 17, 729, 19, 1024, 2187, 2048, 23, 6561, 3125, 8192, 19683, 16384, 29, 59049, 31, 65536, 177147, 131072, 78125, 531441, 37, 524288, 1594323, 1048576, 41, 4782969, 43, 4194304, 14348907

Offset: 1

Author

Amarnath Murthy, Mar 27 2004

Keywords

• nonn

Comments

a(p) = p, a(p*q) = max(p^q, q^p). p,q are primes.

For n>1, maximum among the numbers p^(n/p), where p is a prime factor of n (for minimum, see A243405). Upper bound (for any n): a(n) <= (3^(1/3))^n = A002581^n. - Stanislav Sykora, Jun 04 2014

Examples

			a(12)= 81, the partition into divisors are (12), (6+6),(6+4+2),...(4+4+4), (4+3+3+2), ..., (3+3+3+3), (2+2+2+2+2+2) etc. as 3^4=81 > 4*3*3*2=72 > 2^6 =64.
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A002581, A243405.

Programs

Mathematica

Table[ Max[(n/Divisors[n])^Divisors[n]], {n, 1, 100}] (* Stefan Steinerberger, Apr 23 2006 *)

PARI

A092975(n)={my(p);if(n==1,return(1));
  if(n%3==0,return(3^(n/3)));
  p = factor(n)[1,1];return (p^(n\p));}

Formula

a(n) = Max{(n/d)^d : d divides n }. - Vladeta Jovovic, Aug 06 2005

When n=3m then a(n)=3^m; otherwise, a(n)=q^(n/q), q being the smallest prime factor of n. - Stanislav Sykora, Jun 04 2014

Extensions

More terms from Vladeta Jovovic, Aug 06 2005