|
|
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
|
|
|
FORMULA
|
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
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|