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A361077 a(n) = largest sqrt(2*n)-smooth divisor of binomial(2*n, n).

%I A361077 #34 Dec 07 2024 14:56:47
%S A361077 1,1,2,4,2,36,12,24,18,4,4,24,4,200,5400,720,90,540,300,600,180,120,
%T A361077 120,10800,900,3528,10584,784,280,1680,112,224,882,2940,2940,504,28,
%U A361077 56,4200,19600,980,158760,7560,75600,113400,5040,35280,211680,44100,1800,648,432,216,45360,1680
%N A361077 a(n) = largest sqrt(2*n)-smooth divisor of binomial(2*n, n).
%C A361077 The highest exponent in the prime factorization of a(n) is A263922(n), for n >= 2.
%C A361077 a(n) is even for n >= 2.
%C A361077 Binomial(2*n, n)/a(n) and A263931(n)/a(n) are coprime with a(n) and squarefree. By the Erdős squarefree conjecture, proved in 1996, no a(n) with n >= 5 is squarefree.
%H A361077 Matthieu Pluntz, <a href="/A361077/b361077.txt">Table of n, a(n) for n = 0..10000</a>
%H A361077 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ErdosSquarefreeConjecture.html">Erdős Squarefree Conjecture</a>.
%F A361077 a(n) = binomial(2*n, n) / Product(p prime | p^2 > 2*n, floor(2*n/p) is odd).
%e A361077 binomial(10, 5) = 2^2*3^2*7. 2,3 <= sqrt(10), 7 > sqrt(10) so a(5) = 2^2*3^2 = 36.
%o A361077 (R)
%o A361077 library(primes)
%o A361077 n = 2*10^4
%o A361077 pp = generate_primes(max = sqrt(n))
%o A361077 npinb = list()
%o A361077 for(p in pp){
%o A361077   np = rep(0,n)
%o A361077   for(t in 1:(log(n)/log(p))) np[p*(1:(n/p))] = np[1:(n/p)] + 1 #multiplicities of prime factor p in the integers
%o A361077   npinb[[as.character(p)]] = cumsum(np)[2*(1:(n/2))] - 2*cumsum(np[1:(n/2)]) #multiplicities of prime factor p in the central binomial coefficients
%o A361077 }
%o A361077 res = rep(1,n/2)
%o A361077 for(p in pp){
%o A361077   select = p^2 <= 2*(1:(n/2))
%o A361077   res[select] = res[select]*p^npinb[[as.character(p)]][select]
%o A361077 }
%o A361077 #offset of res is 1
%o A361077 (PARI) a(n) = my(m=sqrtint(2*n), f=factor(binomial(2*n, n), m+1)); for (k=1, #f~, if (f[k,1]>m, f[k,1]=1)); factorback(f); \\ _Michel Marcus_, Mar 03 2023
%Y A361077 Cf. A263922, A263931.
%K A361077 nonn
%O A361077 0,3
%A A361077 _Matthieu Pluntz_, Mar 01 2023