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%I #19 Apr 03 2021 22:01:58
%S 6,9,12,13,14,15,16,18,20,21,22,24,31,32,33,35,39,41,42,43,44,55,56,
%T 57,58,59,60,61,62,65,67,72,73,74,79,107,108,109,110,113,114,115,116,
%U 131,159,219,220,271,319,341,342,1567,1568,1571,1572
%N Numbers k such that the central binomial coefficient C(k, floor(k/2)) has only one prime divisor whose exponent is greater than one.
%C As expected, the (single) non-unitary prime divisors for C(2k, k) and C(k, floor(k/2)) or for Catalan numbers equally come from the smallest prime(s).
%C Numbers k such that A001405(k) is in A190641. - _Michel Marcus_, Jul 30 2017
%C a(56) > 5*10^6 if it exists. - _David A. Corneth_, Apr 03 2021
%e For k=341, binomial(341,170) = 2*2*2*2*M, where M is a squarefree product of 48 further prime factors.
%t pde1Q[n_]:=Length[Select[FactorInteger[Binomial[n,Floor[n/2]]],#[[2]]> 1&]] == 1; Select[Range[1600],pde1Q] (* _Harvey P. Dale_, Jan 21 2019 *)
%o (PARI) isok(n) = my(f=factor(binomial(n, n\2))); #select(x->(x>1), f[,2]) == 1; \\ _Michel Marcus_, Jul 30 2017
%o (PARI) is(n) = { my(nf2 = n\2, nmnf2 = n-nf2, t); forprime(p = 2, n, if(val(n, p) - val(nf2, p) - val(nmnf2, p) > 1, t++; if(t > 1, return(0) ) ) ); t==1 }
%o val(n, p) = my(r=0); while(n, r+=n\=p); r \\ _David A. Corneth_, Apr 03 2021
%Y Cf. A000108, A000984, A001405, A046098, A080664, A081386-A081391, A190641.
%K nonn,more
%O 1,1
%A _Labos Elemer_, Mar 27 2003
%E a(52)-a(55) from _Michel Marcus_, Jul 30 2017
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