OFFSET
1,3
COMMENTS
Probably finite. Erdős believed there were no more terms.
The power of p that divides n! is (n-s)/(p-1), where s is the sum of the "digits" in the base-p representation of n (see Gouvea). - N. J. A. Sloane, Nov 26 2005
From David W. Wilson, Nov 21 2005: (Start)
"The exponent e of prime p in n! can be computed as follows (in pseudo-C):
int e(int p, int n) { int s = 0; while (n > 0) { n /= p; s += n; } return s; }
The exponent of p in C(2n, n) is then f(p, n) = e(p, 2*n) - 2*e(p, n).
We can then use the following function to check if n is in the sequence:
bool isGood(int n) { for (int p = 3; p <= n; p += 2) { if (!isPrime(p)) continue; if (f(p, n) >= 2) return false; } return true; }" (End)
There are no more terms: Granville and Ramaré show that if n >= 2082 then C(2n,n) is divisible by the square of a prime >= sqrt(n/5). - Robert Israel, Sep 18 2017
REFERENCES
F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 100.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, first ed, chap 5, prob 96.
R. K. Guy, Unsolved problems in Number Theory, section B33.
LINKS
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].
EXAMPLE
C(12,6)=924, which is not divisible by the square of an odd prime, so 6 is in the sequence.
MAPLE
filter:= proc(n)
andmap(t -> t[1] = 2 or t[2] = 1, ifactors(binomial(2*n, n))[2]) end proc:
select(filter, [$0..2081]); # Robert Israel, Sep 18 2017
MATHEMATICA
l = {}; Do[m = Binomial[2n, n]; While[EvenQ[m], m = m/2]; If[MoebiusMu[m] != 0, l = {l, n}], {n, 1000}]; Flatten[l] (* Alonso del Arte, Nov 11 2005 *)
(* The following is an implementation of David W. Wilson's algorithm: *)
expoPF[p_, n_] := Module[{s, x}, x = n; s = 0; While[x > 0, x = Floor[x/p]; s = s + x]; s]
expoP2nCn[p_, n_] := expoPF[p, 2*n] - 2*expoPF[p, n]
goodQ[ n_ ] := TrueQ[ Module[ {flag, i}, flag = True; i = 2; While[ flag && Prime[ i ] < n, If[ expoP2nCn[ Prime[ i ], n ] > 1, flag = False ]; i++ ]; flag ] ]
Select[ Range[ 1000 ], goodQ[ # ] & ] (* Alonso del Arte, Nov 11 2005 *)
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Jud McCranie, Jan 01 2001
EXTENSIONS
No other terms for n<=157430. - Naohiro Nomoto, May 12 2002
No other terms for n<=2^31 - 1. - Jack Brennen, Nov 21 2005
STATUS
approved