A267823 -id:A267823 - cp's OEIS frontend

A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

1, 2, 2, 3, 4, 5, 2, 5, 7, 4, 9, 7, 4, 3, 7, 11, 8, 3, 11, 13, 2, 9, 5, 11, 13, 4, 5, 11, 13, 17, 4, 11, 13, 17, 19, 8, 3, 7, 13, 17, 19, 4, 7, 13, 17, 19, 23, 8, 25, 7, 17, 19, 23, 8, 27, 25, 17, 19, 23, 16, 9, 5, 17, 19, 23, 29, 2, 9, 5, 17, 19, 23, 29, 31

Offset: 0

Reinhard Zumkeller, May 25 2013

			.   n        initial rows               A000984(n)   A226047(n)
.  ---+------------------------------+-------------+------------
.   0   [1]                                      1
.   1   [2]                                      2            2
.   2   [2,3]                                    6            3
.   3   [4,5]                                   20            5
.   4   [2,5,7]                                 70            7
.   5   [4,9,7]                                252            9
.   6   [4,3,7,11]                             924           11
.   7   [8,3,11,13]                           3432           13
.   8   [2,9,5,11,13]                        12870           13
.   9   [4,5,11,13,17]                       48620           17
.  10   [4,11,13,17,19]                     184756           19
.  11   [8,3,7,13,17,19]                    705432           19
.  12   [4,7,13,17,19,23]                  2704156           23
.  13   [8,25,7,17,19,23]                 10400600           25
.  14   [8,27,25,17,19,23]                40116600           27
.  15   [16,9,5,17,19,23,29]             155117520           29
.  16   [2,9,5,17,19,23,29,31]           601080390           31
.  17   [4,27,5,11,19,23,29,31]         2333606220           31
.  18   [4,3,25,7,11,19,23,29,31]       9075135300           31
.  19   [8,3,25,7,11,23,29,31,37]      35345263800           37
.  20   [4,9,5,7,11,13,23,29,31,37]   137846528820           37 .

Reinhard Zumkeller, Rows n = 0..250 of triangle, flattened

Cf. A067434 (row lengths), A001316 (left edge), A060308 (right edge), A226047 (row maxima), A226083 (row minima), A000984 (row products).

Cf. A267823.

a226078 n k = a226078_tabf !! n !! k
a226078_row n = a226078_tabf !! n
a226078_tabf = map a141809_row a000984_list

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
b:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      seq(`if`(coeff(p, x, i)>0,
             i^coeff(p, x, i), NULL), i=1..degree(p))
    end:
T:= n-> `if`(n=0, 1, b(n)):
seq(T(n), n=0..30);  # Alois P. Heinz, May 25 2013

Table[Power @@@ FactorInteger[(2n)!/n!^2] , {n, 0, 30}] // Flatten (* Jean-François Alcover, Jul 29 2015 *)

row(n)= if(n<1, [1], [ e[1]^e[2] |e<-Col(factor(binomial(2*n, n)))]); \\ Ruud H.G. van Tol, Nov 18 2024

T(n,k) = A141809(A000984(n),k) for k = 0..A067434(n)-1.

A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

Original entry on oeis.org

3, 10, 10, 3160

Offset: 1

T. D. Noe, Apr 17 2007

The Erdős paper states that it is not known whether the smallest odd prime factor, called g(n), of binomial(2n,n) is bounded. See A129488 for the function g(n). Lucas' Theorem for binomial coefficients can be used to quickly determine whether a prime p divides binomial(2n,n) without computing the binomial coefficient. It is probably a coincidence that 3, 10 and 3160 are all triangular numbers.

Extensive calculations show that if a(5) exists, it is either an integer greater than 13^12 or if it is a triangular number then it is greater than 2^63. [Comment modified by Robert Israel, Jan 27 2016]

			For n=1, binomial(6,3)=20, which is not divisible by 3.
For n=2 and n=3, binomial(20,10)=184756 is not divisible by 3, 5 and 7.
For n=4, binomial(6320,3160), a 1901-digit number, is not divisible by 3, 5, 7 and 11.

P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
Eric Weisstein's World of Mathematics, Lucas Correspondence Theorem

Cf. A000984, A129488, A030979 (n such that g(n)>=11), A266366, A267823.

Table[k = 2; While[AnyTrue[Prime@ Range[2, n + 1], Divisible[Binomial[2 k, k], #] &], k++]; k, {n, 4}] (* Michael De Vlieger, Jan 27 2016, Version 10 *)

isok(kk, n) = {for (j=2, n+1, if (kk % prime(j) == 0, return (0));); return (1);}
a(n) = {my(k = 2); while (! isok(binomial(2*k,k), n), k++); k;} \\ Michel Marcus, Jan 11 2016

a(n) <= A266366(n+1) for n > 0. - Jonathan Sondow, Jan 27 2016

A267825 Index of largest primorial factor of binomial(2n,n).

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 3, 3, 3, 3, 5, 5, 6, 3, 3, 3, 3, 2, 2, 1, 1, 5, 1, 1, 2, 4, 4, 2, 1, 1, 4, 1, 1, 5, 5, 5, 4, 4, 4, 4, 4, 3, 2, 2, 2, 5, 5, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 3, 3, 3, 3, 6, 6, 6, 7, 5, 5, 5, 1, 1, 5, 1, 1, 6, 6, 6, 6, 1, 1, 6, 1, 1, 7, 7, 7, 3, 3, 3

Offset: 0

Jonathan Sondow, Jan 27 2016

nonn

For n > 0, binomial(2n,n) is even, so a(n) >= 1.
Is a(n) unbounded? (The largest value for n <= 100000 is a(45416) = 43.)
From Robert Israel, Jan 28 2016: (Start)
a(n) = A000720(p)-1 where p is the least prime that does not divide A000984(n).
Equivalently, p is the least prime such that the base-p representation of n has all digits < p/2.
a(primorial(k)-1) >= k. In particular the sequence is unbounded. (End)

			Binomial(16,8) = 12870 is divisible by primorial(3) = 2*3*5 = 30, but not by prime(4) = 7, so a(8) = 3.

Wikipedia, Lucas' theorem

Cf. A000720, A000984, A002110, A226078, A267823.

PrimorialFactor[n_] := (k = 0; While[Mod[n, Prime[k + 1]] == 0, k++]; k);
Table[PrimorialFactor[Binomial[2 n, n]], {n, 0, 100}]

pf(n) = {my(k = 0); while (n % prime(k+1) == 0, k++); k;}
a(n) = pf(binomial(2*n, n)); \\ adapted from Mathematica; Michel Marcus, Jan 29 2016

a(A267823(n)) >= n.

min{k : a(k) >= n} = A267823(n).

cp's OEIS Frontend

A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

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A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

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A267825 Index of largest primorial factor of binomial(2n,n).

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