A048570 -id:A048570 - cp's OEIS frontend

A020738 Consider number of divisors of binomial(n, k), k=0..n; a(n) = multiplicity of the maximum value.

2, 1, 2, 1, 2, 1, 4, 3, 4, 1, 4, 3, 2, 1, 2, 1, 6, 2, 6, 2, 6, 1, 8, 2, 2, 1, 4, 2, 2, 1, 10, 4, 2, 5, 2, 2, 2, 1, 2, 1, 6, 2, 2, 2, 4, 1, 2, 1, 2, 2, 6, 2, 4, 2, 2, 4, 2, 1, 10, 2, 2, 3, 4, 8, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 6, 2, 2, 2, 12, 2, 2, 1, 2, 4, 4, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 4, 2

Offset: 1

Labos Elemer

nonn

			If n = 23, the numbers of divisors of {binomial(23, k)} are {1, 2, 4, 8, 16, 16, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 32, ...}. The maximum occurs 8 times, so a(23) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Cf. A000005, A048485, A048569, A048570.

f:= proc(n) local L,k;
  L:= [seq(numtheory:-tau(binomial(n,k)),k=0..n)];
  numboccur(max(L),L)
end proc:
map(f, [$1..200]); # Robert Israel, Nov 17 2016

a[ n_] := If[ n < 1, 0, Last @ Last @ Tally @ Array[ Length @ Divisors @ Binomial[n, #] &, n+1, 0]]; (* Michael Somos, Nov 17 2016 *)

A048475 a(n) is the smallest k at which the number of divisors of binomial(n,k) is maximized.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 5, 4, 5, 6, 7, 7, 8, 6, 7, 6, 7, 7, 11, 8, 11, 11, 13, 12, 13, 14, 15, 11, 13, 13, 14, 15, 17, 18, 19, 19, 20, 18, 19, 20, 21, 19, 23, 23, 24, 23, 24, 20, 23, 22, 23, 24, 23, 28, 29, 21, 29, 23, 23, 29, 21, 31, 29, 29, 24, 31, 29, 24, 29, 29, 31, 31, 33

Offset: 1

Labos Elemer

nonn

			For n=50, the number of divisors of {C(50,k)} is maximal if k=24,26: A000005(C(50,24)) = A000005(C(50,26)) = 5184. The number of divisors of the central (median) value, A000005(C(50,25)) = 4608, is smaller.

Giovanni Resta, Table of n, a(n) for n = 1..4000

Cf. A000005, A048275, A001405, A048570.

a[n_] := Block[{d = DivisorSigma[0, Binomial[n, Range[0, n/2]]]}, Position[ d, Max[d], 1, 1][[1, 1]] - 1]; Array[a, 76] (* Giovanni Resta, May 14 2018 *)

A048569 Values of k for which the number of divisors of the central binomial coefficient C(k, floor(k/2)) exceeds the number of divisors of all other binomial coefficients C(k,j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 13, 14, 15, 16, 22, 26, 29, 30, 37, 38, 39, 40, 46, 47, 48, 57, 58, 85, 86, 87, 93, 94, 95, 97, 98, 106, 107, 122, 123, 124, 125, 147, 148, 149, 150, 157, 158, 159, 178, 194, 206, 214, 219, 220, 226, 230, 232, 247, 278, 283, 284, 285, 286, 316

Offset: 1

Labos Elemer

nonn

k is in the sequence if the number of divisors of the central binomial coefficient C(k, floor(k/2)) (i.e., C(k, k/2) for even k, and C(k,(k-1)/2) = C(k,(k+1)/2) for odd k) is greater than the number of divisors of C(k, j) for all other values of j.

			If n=10 and k=0..10 then A000005(binomial(10,k)) = 1, 4, 6, 16, 16, 18, 16, 16, 6, 4, 1. The maximum value of A000005(binomial(10,k)), i.e., 18 occurs only at k=5, the central coefficient. Thus 10 is in this sequence.

Cf. A000005, A001405, A048274, A034974, A048570.

Edited by Jon E. Schoenfield, May 19 2018

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A020738 Consider number of divisors of binomial(n, k), k=0..n; a(n) = multiplicity of the maximum value.

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A048475 a(n) is the smallest k at which the number of divisors of binomial(n,k) is maximized.

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Mathematica

A048569 Values of k for which the number of divisors of the central binomial coefficient C(k, floor(k/2)) exceeds the number of divisors of all other binomial coefficients C(k,j).

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