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

1,2

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

Links

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

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

Example

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.

Mathematica

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

Prog

(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)); }

Crossrefs

Cf. A002581, A243405.

Sequence in context: A238535 A327415 A072501 * A164340 A046021 A319023

Adjacent sequences: A092972 A092973 A092974 * A092976 A092977 A092978

Keyword

nonn

Author

Amarnath Murthy, Mar 27 2004

Extensions

More terms from Vladeta Jovovic, Aug 06 2005

Status

approved