OFFSET
1,2
COMMENTS
a(n) is the greatest number we may obtain by applying elementary symmetric functions onto the prime factors of n with multiplicity.
The record values of a(n)/n appear at powers of two.
If a(n) is greater than n, then it equals in most cases A003415(n), the first exception where a(n) > A003415(n) > n is at n = 64.
Conjecture equality breaks down after n = 175, as a(A002110(176)) > A024451(176). - Antti Karttunen, Feb 08 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..30030
FORMULA
a(n) = n iff n is not in A083348, otherwise a(n) > n.
a(2^n) = A109388(n) = binomial( n, floor(n/3) )*2^(n-floor(n/3)).
a(p^n) = binomial( n, floor(n/(p+1)) )*p^(n-floor(n/(p+1))), if p is prime.
a(p*n)/a(n) >= n and <= n+1 if p is prime.
a(p*q)/a(q) = p if p and q are prime. This is also true if q is a prime greater than 7 and p is a product of two primes greater than 4.
PROG
(PARI) a(n) = vecmax(Vec(vecprod([(x+f[1])^f[2] | f<-factor(n)~]))) \\ Edited by M. F. Hasler, Feb 14 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Scheuerle, Jul 31 2022
STATUS
approved