A081392 Numbers k such that the central binomial coefficient C(k, floor(k/2)) has only one prime divisor whose exponent is greater than one.

6, 9, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 31, 32, 33, 35, 39, 41, 42, 43, 44, 55, 56, 57, 58, 59, 60, 61, 62, 65, 67, 72, 73, 74, 79, 107, 108, 109, 110, 113, 114, 115, 116, 131, 159, 219, 220, 271, 319, 341, 342, 1567, 1568, 1571, 1572

Offset: 1

Author

Labos Elemer, Mar 27 2003

Keywords

• nonn
• more

Comments

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).

Numbers k such that A001405(k) is in A190641. - Michel Marcus, Jul 30 2017

a(56) > 5*10^6 if it exists. - David A. Corneth, Apr 03 2021

Examples

			For k=341, binomial(341,170) = 2*2*2*2*M, where M is a squarefree product of 48 further prime factors.
		

Crossrefs

Cf. A000108, A000984, A001405, A046098, A080664, A081386-A081391, A190641.

Programs

Mathematica

pde1Q[n_]:=Length[Select[FactorInteger[Binomial[n,Floor[n/2]]],#[[2]]> 1&]] == 1; Select[Range[1600],pde1Q] (* Harvey P. Dale, Jan 21 2019 *)

PARI

isok(n) = my(f=factor(binomial(n, n\2))); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017

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 }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Apr 03 2021

Extensions

a(52)-a(55) from Michel Marcus, Jul 30 2017