A067315 -id:A067315 - cp's OEIS frontend

A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

2, 12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 420, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 920, 936, 952, 966, 986, 992, 1020, 1054, 1102

Offset: 1

Dean Hickerson, Jan 16 2002

nonn

This sequence has a surprisingly large overlap with A080385(n); a few values, 2, 420, 920 are exceptional. This means that usually A080383(A067348(n))=7. - Labos Elemer, Mar 17 2003

Conjecture: sequence contains most of 2*A000384(k). Exceptions are k = 8, 18, 20, 23, 35, ... - Ralf Stephan, Mar 15 2004

David A. Corneth, Table of n, a(n) for n = 1..12000

Subsequence of A042996.

Cf. A000984, A067315, A080385.

Select[Range[2, 1200, 2], Mod[Binomial[ #, #/2], # ]==0&]

val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {if(valuation(n, 2) == 0, return(0)); my(f = factor(n)); for(i=1, #f~, if(val(n, f[i, 1]) - 2 * val(n/2, f[i, 1]) - f[i, 2] < 0, return(0))); return(1)} \\ David A. Corneth, Jul 29 2017

a(n) = 2*A002503(n-2) + 2.

Appears to be 2*A058008(n). - Benoit Cloitre, Mar 21 2003

Name clarified by Peter Luschny, Aug 04 2017

A042996 Numbers k such that binomial(k, floor(k/2)) is divisible by k.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 56, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121

Offset: 1

Labos Elemer

nonn

All the odd numbers are terms. - Amiram Eldar, Aug 24 2024

			For n = 12, binomial(12,6) = 924 = 12*77 is divisible by 12, so 12 is in the sequence.
For n = 13, binomial(13,6) = 1716 = 13*132 is divisible by 13, so 13 is in the sequence.
From _David A. Corneth_, Apr 22 2018: (Start)
For n = 20, we wonder if 20 = 2^2 * 5 divides binomial(20, 10) = 20! / (10!)^2.
The exponent of 2 in the prime factorization of 20! is 10 + 5 + 2 + 1 = 18.
The exponent of 2 in the prime factorization of 10! is 5 + 2 + 1 = 8.
Therefore, the exponent of 2 in binomial(20, 10) is 18 - 2*8 = 2.
The exponent of 5 in the prime factorization of 20! is 4.
The exponent of 5 in the prime factorization of 10! is 2.
Therefore, exponent of 5 in binomial(20, 10) is 4 - 2*2 = 0.
So binomial(20, 10) is not divisible by 20, and 20 is not in the sequence. (End)

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001405, A020475, A067315 (complement).

Select[Range[150],Divisible[Binomial[#,Floor[#/2]],#]&] (* Harvey P. Dale, Sep 15 2011 *)

isok(n) = (binomial(n, n\2) % n) == 0; \\ Michel Marcus, Apr 22 2018

A215619 a(n) is the number of consecutive terms of A100071, beginning with index n, which are divisible by n.

Original entry on oeis.org

4, 1, 6, 1, 8, 1, 4, 1, 12, 5, 14, 1, 4, 1, 18, 1, 20, 1, 4, 1, 24, 1, 6, 1, 4, 1, 30, 21, 32, 1, 12, 1, 8, 1, 38, 1, 14, 1, 42, 1, 44, 1, 6, 1, 48, 1, 8, 1, 4, 1, 54, 1, 6, 9, 4, 1, 60, 1, 62, 1, 4, 1, 6, 1, 68, 1, 4, 1, 72, 1, 74, 1, 4, 1, 12, 1, 80, 1, 4, 1

Offset: 3

Vladimir Shevelev, Aug 17 2012

nonn

a(n) = n+1 iff n is prime.
a(n) = 1  iff n in { A067315 }.
1 <= a(n) <= n+1.
{ n : a(2n)>1 } = { A058008 } \ { 1 }.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Cf. A000040, A067315, A100071.

b:= proc(n) b(n):= n * binomial(n-1, floor((n-1)/2)) end:
a:= proc(n) local k;
      for k from 0 while irem(b(n+k), n)=0 do od; k
    end:
seq (a(n), n=3..100);  # Alois P. Heinz, Aug 17 2012

b[n_] := n Binomial[n-1, Floor[(n-1)/2]];
a[n_] := Module[{k = 0}, While[Mod[b[n+k], n] == 0, k++]; k];
a /@ Range[3, 100] (* Jean-François Alcover, Nov 22 2020, after Alois P. Heinz *)

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A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

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A042996 Numbers k such that binomial(k, floor(k/2)) is divisible by k.

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A215619 a(n) is the number of consecutive terms of A100071, beginning with index n, which are divisible by n.

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