A048484 -id:A048484 - cp's OEIS frontend

A020731 Numbers n for which number of distinct prime divisors of C(n,k) has maximum at k = [n/2].

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 47, 48, 49, 50, 55, 56, 57, 58, 63, 64, 65, 66, 67, 68, 71, 72, 73, 75, 76, 80, 83, 84, 85, 89, 90, 96, 97, 98, 99, 100, 107, 108, 109, 119

Offset: 1

Labos Elemer

nonn

			For n=21 the number of prime divisors of {C(21,k)} is {0,2,4,4,4,4,5,5,6,6,6,6,6,6,5,5,4,4,4,4,2,0}, the maximal value of 6 occurring at the central position.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A001221, A048486, A048299, A048484, A019491.

Select[Range[120], Function[n, ar = PrimeNu[Binomial[n, Range[0, n/2]]]; Max[ar] == ar[[-1]]]] (* Ivan Neretin, Aug 14 2015 *)

A019491 Numbers n for which number of distinct prime divisors of binomial(n,k) has local minimum at k = n/2.

Original entry on oeis.org

10, 20, 27, 28, 29, 34, 38, 44, 45, 46, 51, 52, 53, 54, 60, 61, 62, 69, 70, 74, 77, 78, 79, 81, 82, 87, 88, 92, 93, 94, 95, 101, 102, 103, 104, 105, 106, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 122, 124, 125, 126, 127, 130, 138, 139, 140

Offset: 1

Labos Elemer

nonn

			If n=28 then {r(C(28,k))}={0,2,3,4,4,5,6,6,6,7,8,7,7,7,6,7,7,7,8,7,6,6,6,5,4,4,3,2,0}. Thus r(C(28,14))=6 is local minimum, while r(C(28,10))=8 is maximum.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A048484, A048486, A001221, A020731.

Select[Range[140], MatchQ[PrimeNu[Binomial[#, Range[Floor[#/2], #]]], {(x_) .., y_, _} /; x < y]&] (* Jean-François Alcover, Dec 10 2016 *)

Data corrected by Jean-François Alcover, Dec 10 2016

A020733 Consider number of prime divisors of binomial(n,k), k=0..n; a(n) = multiplicity of the maximum value.

Original entry on oeis.org

2, 1, 2, 1, 2, 5, 4, 1, 4, 2, 4, 1, 2, 5, 8, 1, 2, 5, 8, 2, 6, 7, 8, 5, 8, 11, 2, 2, 4, 11, 10, 3, 8, 2, 6, 3, 6, 2, 4, 1, 2, 5, 8, 2, 12, 16, 16, 5, 6, 13, 8, 12, 12, 4, 8, 5, 4, 5, 6, 4, 2, 6, 10, 1, 2, 7, 6, 5, 2, 2, 12, 15, 16, 2, 8, 11, 2, 10, 10, 11, 2, 6, 12, 3, 16, 2, 4, 8, 10, 5, 2, 2, 4, 6

Offset: 1

Labos Elemer

nonn

			The number of distinct primes of binomial(15,k) are {0,2,3,3,4,4,4,4,4,4,4,4,3,3,2,0}. The maximum is 4 and it occurs 8 times, thus a(15) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Robert Israel)

Cf. A001221, A048484, A048486.

f:= proc(n) local A,i;
  A:= [seq(nops(numtheory:-factorset(binomial(n,i))),i=0..n)];
  numboccur(max(A),A);
end proc:
map(f, [$1..100]); # Robert Israel, May 26 2020

a[n_] := Sort[Tally[Table[PrimeNu[Binomial[n, k]], {k, 0, n}]]][[-1, 2]];
Array[a, 100] (* Jean-François Alcover, Jun 09 2020 *)

a(n) = {v = vector(n+1, k, omega(binomial(n, k-1))); m = vecmax(v); sum(i=1, n+1, v[i] == m);} \\ Michel Marcus, Dec 30 2013

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A020731 Numbers n for which number of distinct prime divisors of C(n,k) has maximum at k = [n/2].

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A019491 Numbers n for which number of distinct prime divisors of binomial(n,k) has local minimum at k = n/2.

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A020733 Consider number of prime divisors of binomial(n,k), k=0..n; a(n) = multiplicity of the maximum value.

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