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

1,1

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

Links

Table of n, a(n) for n=1..55.

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A143710 A087022 A315951 * A225868 A185177 A185179

Adjacent sequences: A081389 A081390 A081391 * A081393 A081394 A081395

Keyword

nonn,more

Author

Labos Elemer, Mar 27 2003

Extensions

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

Status

approved