

A065352


Smallest m such that C(2m,m) is divisible by (m+n)!/m!.


1



1, 3, 8, 19, 42, 153, 216, 375, 950, 3565, 4068, 12273, 12274, 31729, 122352, 131023, 458222, 522221, 1046508, 3145451, 6291178, 12320745, 16769000, 56623079, 113246182
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OFFSET

1,2


COMMENTS

For n=1 see Catalan numbers A000108.
Heuristically one can observe that a(n) + n has a 'high' valuation of 2. For n = 17..25 we have 2^8(a(n) + n + 1). Using this heuristic we find a(26)..a(31) <= 267780069, 469745636, 671088611, 1879015394, 2146959329 and 6442418144 respectively.  David A. Corneth, Mar 28 2021


LINKS



FORMULA

C(2m, m)=A*((m+1)(m+2)...(m+n1)(m+n)); a(n) is the smallest such m belonging to n: a(n)=Min(m; Mod(A000984(m), (m+n)!/m!)=0)


EXAMPLE

n=4: a(4)=19 means that C(38,19)=35345263800 is divisible by (19+1)(19+2)(19+3)(19+4)=23!/19!=20*21*22*23=215520; the quotient is 166315. Smaller (<19) central binomial coefficients are not divisible by such a product of 4 successive terms; the corresponding quotients for n = 1, 2, 3, 4, 5,... are 1, 1, 13, 166315, 9120910752273999,...


MATHEMATICA

Do[m = 1; While[Not[Divisible[Binomial[2*m, m], (m+n)!/m!]], m++]; Print[m], {n, 1, 16}] (* Vaclav Kotesovec, Sep 05 2019 *)


PROG

(PARI) \\ See Corneth link


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



