

A356253


a(n) is the largest coefficient of P(x) := Product_{k} (x + p_k), where (p_k) are the primes dividing n listed with multiplicity.


3



1, 2, 3, 4, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 32, 17, 21, 19, 24, 21, 22, 23, 44, 25, 26, 27, 32, 29, 31, 31, 80, 33, 34, 35, 60, 37, 38, 39, 68, 41, 42, 43, 48, 45, 46, 47, 112, 49, 50, 51, 56, 53, 81, 55, 92, 57, 58, 59, 92, 61, 62, 63, 240, 65, 66, 67, 72
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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.


LINKS



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^(nfloor(n/3)).
a(p^n) = binomial( n, floor(n/(p+1)) )*p^(nfloor(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

Cf. A065048 (same concept but uses numbers 1..n instead of prime factors of n).


KEYWORD

nonn


AUTHOR



STATUS

approved



