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A092975 Consider all partitions of n into parts all of which are divisors of n; a(n) = maximal product of parts. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 29 01:22 EDT 2021. Contains 346340 sequences. (Running on oeis4.)