1,3

Probably finite. Erdos 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). - njas, Nov 26, 2005

Comments from David W. Wilson, Nov 21 2005: "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; }"

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.

C(12,6)=924, which is not divisible by the square of an odd prime, so 6 is in the sequence.

l = {}; Do[m = Binomial[2n, n]; While[EvenQ[m], m = m/2]; If[MoebiusMu[m] != 0, l = {l, n}], {n, 1000}]; Flatten[l] (Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 11 2005)

Alonso Delarte (alonso.delarte(AT)gmail.com), Nov 11 2005: The following is an implementation of David W. Wilson's algorithm in Mma:

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[ # ] & ]

Cf. A000984. Cf. also A081391, n such that C[2n, n] has only one prime divisor of which exponent equals one.

Sequence in context: A047519 A070118 A070124 this_sequence A047301 A001953 A079057

Adjacent sequences: A059094 A059095 A059096 this_sequence A059098 A059099 A059100

nonn,fini

Jud McCranie (j.mccranie(AT)comcast.net), Jan 01 2001

No other terms for n<=157430. - Naohiro Nomoto (n_nomoto(AT)yabumi.com), May 12 2002

No other terms for n<=2^31 - 1. - Jack Brennen (jb(AT)brennen.net), Nov 21 2005