cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, May 25 2013

Keywords

Examples

			.   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 .

Links

Crossrefs

Cf. A067434 (row lengths), A001316 (left edge), A060308 (right edge), A226047 (row maxima), A226083 (row minima), A000984 (row products).
Cf. A267823.

Programs

  • Haskell
    a226078 n k = a226078_tabf !! n !! k
a226078_row n = a226078_tabf !! n
a226078_tabf = map a141809_row a000984_list
  • Maple
    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
  • Mathematica
    Table[Power @@@ FactorInteger[(2n)!/n!^2] , {n, 0, 30}] // Flatten (* Jean-François Alcover, Jul 29 2015 *)
  • PARI
    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

Formula

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

A226083 Smallest element of the set of largest prime powers p^k dividing C(2*n,n), where p is any prime factor of C(2*n,n).

Original entry on oeis.org

2, 2, 4, 2, 4, 3, 3, 2, 4, 4, 3, 4, 7, 8, 5, 2, 4, 3, 3, 4, 3, 3, 13, 4, 8, 8, 16, 5, 3, 7, 7, 2, 3, 3, 7, 4, 7, 3, 11, 4, 5, 5, 7, 7, 5, 5, 5, 4, 8, 8, 11, 8, 5, 3, 3, 8, 3, 3, 5, 7, 7, 7, 3, 2, 4, 3, 3, 4, 7, 8, 11, 4, 8, 8, 5, 5, 5, 7, 7, 4, 5, 5, 3, 7, 5, 5, 3, 3, 9, 11, 7, 3, 7, 7, 13, 4, 8, 8, 3, 3
Offset: 1

Views

Author

Alois P. Heinz, May 25 2013

Keywords

Examples

			a(89) = 9: C(2*89,89) = 2^4 * 3^2 * 5^3 * 7^2 * 11^1 * ... * 173^1, the smallest prime power is 3^2 = 9.  3^2 is the largest prime power for prime 3 dividing C(2*89,89).
a(9993) = 59: 59^1 is the largest power of 59 dividing C(2*9993,9993), it is smaller than the largest powers of all other prime factors.

Links

Crossrefs

Cf. A000040, A000961, A000984, A007318, A226047 (row maxima of A226078).

Programs

  • Maple
    a:= proc(n) local h, i, m, p;
      p:=1; m:=infinity;
      while p < m do p:= nextprime(p); i:= 0;
         h:= 2*n; while h>0 do h:=iquo(h, p); i:=i+h od;
         h:= n;   while h>0 do h:=iquo(h, p); i:=i-2*h od;
         if i>0 then m:= min(m, p^i) fi
      od; m
    end:
seq(a(n), n=1..100);

Formula

a(n) = min_{p prime, p|C(2n,n)} max_{k, p^k|C(2n,n)} p^k.
a(n) = min_{k=0..A067434(n)-1} A226078(n,k).

A227902 Numbers n such that triangular(n) divides binomial(2n,n).

Original entry on oeis.org

1, 2, 4, 6, 15, 20, 24, 28, 40, 42, 45, 66, 72, 77, 88, 91, 104, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 240, 266, 276, 299, 304, 308, 312, 315, 322, 325, 330, 345, 368, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 464, 468, 476, 480
Offset: 1

Views

Author

Alex Ratushnyak, Oct 14 2013

Keywords

Comments

A014847 is a subsequence.

Examples

			triangular(6)=21, A000984(6)=924. Because 21 divides 924, 6 is in the sequence.

Links

Crossrefs

Cf. A000984, A014847, A081766, A081767, A110493-A110496, A226047.

Programs

  • Mathematica
    Select[Range[480], Mod[Binomial[2 #, #], # (# + 1)/2] == 0 &] (* T. D. Noe, Oct 16 2013 *)
  • PARI
    is(n) = { my(f = factor(binomial(n+1, 2))); for(i = 1, #f~, if(val(2*n, f[i, 1]) - 2*val(n, f[i, 1]) < f[i, 2], return(0) ) ); 1 }
val(n, p) = my(r=0); while(n, r+=n\=p);r \\ David A. Corneth, Apr 03 2021
  • Python
    from sympy import binomial
for n in range(1, 444):
    CBC = binomial(2 * n, n)
    if not CBC % binomial(n + 1, 2):
       print(n, end=",")
Showing 1-3 of 3 results.

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