A020733 -id:A020733 - cp's OEIS frontend

A048685 a(n) is the number of times the maximum value of Omega(binomial(n, k)) occurs in the n-th row of Pascal's triangle, where Omega(n) is the number of prime divisors of n counted with multiplicity (A001222).

2, 1, 2, 3, 2, 1, 4, 2, 6, 3, 4, 2, 2, 1, 2, 4, 4, 2, 8, 2, 2, 3, 8, 2, 2, 3, 2, 3, 2, 1, 2, 4, 2, 4, 8, 2, 2, 2, 12, 2, 6, 2, 2, 2, 2, 1, 2, 2, 2, 4, 8, 2, 4, 2, 4, 2, 2, 5, 4, 2, 4, 1, 18, 2, 8, 2, 2, 10, 8, 2, 2, 6, 2, 2, 2, 2, 4, 2, 10, 8, 4, 4, 2, 6, 2, 3, 2, 2, 4, 2, 2, 2, 2, 1, 2, 4, 4, 2, 4, 2, 4, 4

Offset: 1

Labos Elemer

nonn

			For n = 19, the A001222 spectrum for binomial(n,k) is: {0, 1, 3, 3, 5, 6, 6, 6, 6, 5, 5, 6, 6, 6, 6, 5, 3, 3, 1, 0}. The maximum arises 8 times, so a(19) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A007318, A020738, A020733.

a[n_] := Module[{row = Table[PrimeOmega[Binomial[n, k]], {k, 0, n}]}, Count[row, Max[row]]]; Array[a, 100] (* Amiram Eldar, Aug 13 2024 *)

A335209 Numbers k such that binomial(2k,k) has more distinct prime factors than binomial(2k,i) for 0 <= i < k.

Original entry on oeis.org

1, 2, 4, 6, 8, 20, 32, 54, 66, 110, 144, 170, 200, 210, 278, 288, 304, 330, 402, 405, 468, 510, 527, 628, 654, 684, 704, 778, 783, 784, 853, 891, 892, 990, 1001, 1035, 1125, 1155, 1232, 1296, 1334, 1384, 1394, 1488, 1495, 1521, 1551, 1575, 1600, 1625, 1645, 1701, 1768, 1875, 1891, 2028, 2072

Offset: 1

Robert Israel, May 26 2020

nonn

Numbers k such that A020733(2*k) = 1.

			a(4)=6 is in the sequence because binomial(12,6) = 924 = 2^2*3*7*11 has 4 distinct prime factors while binomial(12,0) to binomial(12,5) all have at most 3.
7 is not in the sequence because binomial(14,7) = 3432 = 2^3*3*11*13 and binomial(14,6) = 3003 = 3*7*11*13 both have 4 distinct prime factors.

Robert Israel, Table of n, a(n) for n = 1..250

Cf. A001221, A020733, A067434.

filter:=  proc(n) local t, v, i, m;
  m:= 0: t:= 1:
  for i from 1 to n-1 do
   t:= t * ifactor(2*n-i+1)/ifactor(i);
   if type(t,`*`) then v:= nops(t) else v:= 1 fi;
   if v > m then m:= v fi;
  od;
  t:= t*ifactor(n+1)/ifactor(n);
  type(t,`*`) and nops(t) > m
end proc:
filter(1):= true:
select(filter, [$1..2500]); # Robert Israel, May 26 2020

Select[Range@ 1001, Max@ Most@ # < Last@ # &@ PrimeNu@ Binomial[2 #, Range[0, #]] &] (* Michael De Vlieger, May 26 2020 *)

cp's OEIS Frontend

A048685 a(n) is the number of times the maximum value of Omega(binomial(n, k)) occurs in the n-th row of Pascal's triangle, where Omega(n) is the number of prime divisors of n counted with multiplicity (A001222).

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A335209 Numbers k such that binomial(2k,k) has more distinct prime factors than binomial(2k,i) for 0 <= i < k.

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cp's OEIS Frontend

A048685 a(n) is the number of times the maximum value of Omega(binomial(n, k)) occurs in the n-th row of Pascal's triangle, where Omega(n) is the number of prime divisors of n counted with multiplicity (A001222).

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Mathematica

A335209 Numbers k such that binomial(2*k,k) has more distinct prime factors than binomial(2*k,i) for 0 <= i < k.

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Maple

Mathematica

A335209 Numbers k such that binomial(2k,k) has more distinct prime factors than binomial(2k,i) for 0 <= i < k.