A356253 - OEIS

%I #45 Feb 14 2024 14:24:00

%S 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,

%T 26,27,32,29,31,31,80,33,34,35,60,37,38,39,68,41,42,43,48,45,46,47,

%U 112,49,50,51,56,53,81,55,92,57,58,59,92,61,62,63,240,65,66,67,72

%N 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.

%C a(n) is the greatest number we may obtain by applying elementary symmetric functions onto the prime factors of n with multiplicity.

%C The record values of a(n)/n appear at powers of two.

%C 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.

%C Conjectured: a(A002110(n)) = A024451(n), for n > 2.

%C Conjecture equality breaks down after n = 175, as a(A002110(176)) > A024451(176). - _Antti Karttunen_, Feb 08 2024

%H Antti Karttunen, <a href="/A356253/b356253.txt">Table of n, a(n) for n = 1..30030</a>

%F a(n) = n iff n is not in A083348, otherwise a(n) > n.

%F a(2^n) = A109388(n) = binomial( n, floor(n/3) )*2^(n-floor(n/3)).

%F a(p^n) = binomial( n, floor(n/(p+1)) )*p^(n-floor(n/(p+1))), if p is prime.

%F a(p*n)/a(n) >= n and <= n+1 if p is prime.

%F 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.

%F a(A002110(n)) >= A024451(n), for n > 2. The maximum of row n in A260613 a variant of A070918.

%o (PARI) a(n) = vecmax(Vec(vecprod([(x+f[1])^f[2] | f<-factor(n)~]))) \\ Edited by _M. F. Hasler_, Feb 14 2024

%Y Cf. A002110, A003415, A024451, A070918, A083348, A109388, A260613, A369657 (difference between this sequence and A003415).

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

%K nonn

%O 1,2

%A _Thomas Scheuerle_, Jul 31 2022