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 1-4 of 4 results.

A260641 Numbers n such that 2^n-1 is in A004782 but not in A081767.

Original entry on oeis.org

2, 3, 7, 9, 12, 13, 17, 23, 25, 26, 31, 33, 37, 41, 44, 45, 46, 48, 49, 55, 56, 57, 61, 65, 67
Offset: 1

Views

Author

M. F. Hasler, Nov 11 2015

Keywords

Comments

It appears that the numbers in A004782 \ A081767 are all of the form 2^n-1; this sequence gives the corresponding n-values.

Links

Programs

  • PARI
    for(k=2,999, n=2^k-1; !binomod(2*n-2, n-1, n^2-n) && binomod(2*n, n,n^2-1) && print1(k", "))

A260642 Numbers in A004782, n-1 | C(n-1), but not in A081767, n-1 | C(n), where C(n) = A000108(n) = 2n!/n!(n+1)! are the Catalan numbers.

Original entry on oeis.org

3, 7, 127, 511, 4095, 8191, 131071, 8388607, 33554431, 67108863, 2147483647, 8589934591, 137438953471, 2199023255551, 17592186044415, 35184372088831, 70368744177663, 281474976710655, 562949953421311, 36028797018963967, 72057594037927935
Offset: 1

Views

Author

M. F. Hasler, Nov 11 2015

Keywords

Comments

It appears that all terms are of the form 2^k-1 ; see A260641 for the k-values.

Crossrefs

Cf. A000108, A260641, A004782, A081767, A014847.

Formula

a(n) = 2^A260641(n)-1 = A000225(A260641(n)), i.e., equals A000225 o A260641.
Equals A004782 \ A081767.

A014847 Numbers k such that k-th Catalan number C(2k,k)/(k+1) is divisible by k.

Original entry on oeis.org

1, 2, 6, 15, 20, 28, 42, 45, 66, 77, 88, 91, 104, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 240, 266, 276, 299, 308, 312, 315, 322, 325, 330, 345, 368, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 464, 468, 476, 483, 493
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The sequence does not contain any odd primes p (follows by quadratic reciprocity and field structure of Z/pZ). Aside from the first 2 terms, all other terms are composite integers. - Thomas M. Bridge, Nov 03 2013
Equivalently, numbers such that binomial(2n, n) = 0 (mod n). Indices of zeros in A059288. See A260640 (and A260636) for the analogs for 3n. - M. F. Hasler, Nov 11 2015
The 2nd comment is true because gcd(n,n+1) = 1 and n+1 divides C(2n,n). The 1st comment then follows, because prime p does not divide C(2p,p) = 2p*(2p-1)*...*(p+1)/(p*(p-1)*...*1) unless p = 2. - Jonathan Sondow, Jan 07 2018
A number n is in the sequence if and only if, for each prime p dividing n, the number of carries in the addition n+n in base p is at least the p-adic valuation of n. In particular, if n is squarefree, the condition is that at least one base-p digit of n is at least p/2. - Robert Israel, Jan 07 2018
If A is the set of all a(k)'s, Pomerance proved that the upper density of A is at most 1 - log 2 = 0.30685... and conjectured that A has positive lower density. I improved Pomerance's result by showing that the upper density of A is at most 1 - log 2 - 0.05551 = 0.25134... Numerically, this upper density seems to be less than 0.11. - Carlo Sanna, Jan 28 2018
The asymptotic density of this sequence is 0.11424743... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

Links

Crossrefs

Cf. A000108, A000984, A120622, A120623, A120624, A120625, A120626, A121943, A282163, A282346, A283073, A283074, A282672.

Programs

  • Magma
    [n: n in [1..500] | IsZero((Binomial(2*n, n) div (n+1)) mod n)]; // Vincenzo Librandi, Jan 29 2016
  • Maple
    filter:= proc(n) local F, f, r, c, t,j;
  F:= ifactors(n)[2];
  for f in F do
    r:= convert(n,base,f[1]);
    c:= 0: t:= 0:
    for j from 1 to nops(r) do
      if 2*r[j]+c >= f[1] then
          c:= 1; t:= t+1;
      else c:= 0
      fi;
    od;
    if t < f[2] then return false fi;
  od;
  true
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 07 2018
  • Mathematica
    fQ[n_] := IntegerQ[Binomial[2n, n]/ n]; Select[ Range@495, fQ@# &] (* Robert G. Wilson v, Jun 19 2006 *)
Select[Table[{CatalanNumber[n],n},{n,500}],Divisible[#[[1]],#[[2]]]&][[All,2]] (* Harvey P. Dale, Nov 07 2022 *)
  • PARI
    is_A014847(n)=!binomod(2*n,n,n) \\ Suitable for large n. Using binomod.gp by M. Alekseyev, cf. links. - M. F. Hasler, Nov 11 2015
  • PARI
    for(n=1, 1e3, if(binomial(2*n, n)/(n+1) % n==0, print1(n", "))) \\ Altug Alkan, Nov 11 2015
  • Python
    from _future_ import division
A014847_list, b = [], 1
for n in range(1,10**3):
    if not b % n:
        A014847_list.append(n)
    b = b*(4*n+2)//(n+2) # Chai Wah Wu, Jan 27 2016

Formula

It seems that a(n)/n is bounded and more precisely that lim_{n -> infinity} a(n)/n = C exists with 9 <= c < 10. - Benoit Cloitre, Aug 13 2002
a(n) = A004782(n) - 1. - Enrique Pérez Herrero, Feb 03 2013

A081767 Numbers k such that k^2 - 1 divides binomial(2k,k).

Original entry on oeis.org

2, 16, 21, 29, 43, 46, 67, 78, 89, 92, 105, 111, 141, 154, 157, 171, 188, 191, 205, 210, 211, 221, 229, 232, 239, 241, 267, 277, 300, 309, 313, 316, 323, 326, 331, 346, 369, 379, 415, 421, 430, 436, 441, 443, 451, 460, 461, 465, 469, 477, 484, 494, 497, 528
Offset: 1

Views

Author

Benoit Cloitre, Apr 09 2003

Keywords

Comments

Is a(n) asymptotic to c*n with 9 < c < 10?
A subset of A004782: numbers k such that 2(2k-3)!/(k!(k-1)!) is an integer.
Equivalently, numbers k such that k-1 divides A000108(k), the k-th Catalan number. - M. F. Hasler, Nov 11 2015
The data does not appear to support the conjectured asymptote statement (neither the constant nor being linear). - Bill McEachen, Feb 26 2024

Links

Crossrefs

Subsequence of A094575 and of A004782.
Cf. A000108.

Programs

  • Mathematica
    Select[Range[2,600],Divisible[Binomial[2#,#],#^2-1]&] (* Harvey P. Dale, May 11 2013 *)
  • PARI
    for(n=2, 999, binomial(2*n, n)%(n^2-1)||print1(n", ")) \\ M. F. Hasler, Nov 11 2015
  • PARI
    is_A081767(n)=!binomod(2*n, n, n^2-1) \\ Using binomod.gp by Max Alekseyev, cf. links. - M. F. Hasler, Nov 11 2015
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.