A034974 -id:A034974 - cp's OEIS frontend

A034973 Number of distinct prime factors in central binomial coefficients C(n, floor(n/2)), the terms of A001405.

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 13, 13, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 14, 14, 15, 15, 15, 15, 16

Offset: 1

Labos Elemer

Sequence is not monotonic. E.g., a(44)=10, a(45)=9 and a(46)=10. The number of prime factors of n! is pi(n), but these numbers are lower.

Prime factors are counted without multiplicity. - Harvey P. Dale, May 20 2012

			a(25) = omega(binomial(25,12)) = omega(5200300) = 6 because the prime factors are 2, 5, 7, 17, 19, 23.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A001405, A034974, A067434.

Table[PrimeNu[Binomial[n,Floor[n/2]]],{n,90}] (* Harvey P. Dale, May 20 2012 *)

a(n)=omega(binomial(n,n\2)) \\ Charles R Greathouse IV, Apr 29 2015

A056059 GCD of largest square and squarefree part of central binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 6, 2, 1, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 6, 3, 1, 1, 1, 2, 3, 6, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 3, 6, 6, 3, 1, 2, 2, 1, 2, 1, 3, 6, 1, 1, 1, 2, 1, 2

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, squarefree part is 858. So a(14) = gcd(4,858) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000188, A001405, A007913, A008833, A034974, A046098, A055229, A055231.
Cf. A056056, A056057, A056058, A056060, A056061.
A056201 is the cube of this sequence.

Table[GCD[First@ Select[Reverse@ Divisors@ #, IntegerQ@ Sqrt@ # &], Times @@ Power @@@ Map[{#1, Mod[#2, 2]} & @@ # &, FactorInteger@ #]] &@ Binomial[n, Floor[n/2]], {n, 80}] (* Michael De Vlieger, Feb 18 2017, after Zak Seidov at A007913 *)

A001405(n) = binomial(n, n\2);
A055229(n) = { my(c=core(n)); gcd(c, n/c); } \\ Charles R Greathouse IV, Nov 20 2012
A056059(n) = A055229(A001405(n)); \\ Antti Karttunen, Jul 20 2017

from sympy import binomial, gcd
from sympy.ntheory.factor_ import core
def a001405(n): return binomial(n, n//2)
def a055229(n):
    c = core(n)
    return gcd(c, n//c)
def a(n): return a055229(a001405(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 20 2017

a(n) = A055229(A001405(n)), where A055229(n) = gcd(A008833(n), A007913(n)).

Formula clarified by Antti Karttunen, Jul 20 2017

A056057 The largest square which divides n-th central binomial coefficient.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 9, 36, 1, 4, 4, 4, 9, 9, 1, 4, 1, 4, 4, 4, 1, 4, 100, 100, 900, 900, 36, 144, 9, 9, 9, 36, 25, 100, 100, 100, 9, 36, 4, 4, 4, 4, 900, 3600, 225, 900, 1764, 1764, 1764, 1764, 196, 784, 4, 4, 4, 16, 4, 16, 16, 16, 441, 441, 49, 196, 49, 196, 36, 36, 1, 4

Offset: 1

Labos Elemer, Jul 26 2000

nonn

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

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974.

Cf. A056056, A056058, A056059, A056060, A056061.

Table[First@ Select[Reverse@ Divisors@ Binomial[n, Floor[n/2]], IntegerQ@ Sqrt@ # &], {n, 72}] (* Michael De Vlieger, Feb 18 2017 *)
a[n_] := Times @@ (First[#]^(2*Floor[Last[#]/2]) & /@ FactorInteger[Binomial[n, Floor[n/2]]]); Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

a(n) = A008833(A001405(n)).

a(A046098(n)) = 1.

A056060 The powerfree part of the central binomial coefficients.

Original entry on oeis.org

1, 2, 3, 6, 10, 5, 35, 70, 14, 7, 462, 231, 429, 429, 715, 1430, 24310, 12155, 92378, 46189, 88179, 88179, 1352078, 676039, 52003, 52003, 7429, 7429, 1077205, 1077205, 33393355, 66786710, 43214930, 21607465, 181502706, 90751353, 176726319, 176726319, 7658140490

Offset: 1

Labos Elemer, Jul 26 2000

nonn

			n=14, binomial(14,7) = 3432 = 8*3*11*13. The largest square divisor is 4, and the squarefree part is 858. So GCD(4,858) = 2 and a(14) = 858/2 = 429.

Amiram Eldar, Table of n, a(n) for n = 1..3388

Cf. A001405, A000188, A008833, A007913, A055229, A055231, A046098, A034974, A056056, A056057, A056058, A056059, A056060, A056061.

a[n_] := Denominator[(b = Binomial[n, Floor[n/2]])/(Times @@ First /@ FactorInteger[b])^2]; Array[a, 36] (* Amiram Eldar, Sep 05 2020 *)

a(n) = A055231(A001405(n)).

New name and more terms from Amiram Eldar, Sep 05 2020

A048275 a(n) = maximal value for number of divisors of C(n,k) for k=0..n.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 4, 8, 12, 18, 16, 24, 24, 32, 24, 48, 32, 60, 48, 80, 96, 128, 64, 160, 192, 192, 288, 480, 384, 480, 192, 576, 768, 768, 1024, 1536, 1152, 1536, 1536, 2304, 1536, 2880, 2304, 2560, 3072, 3840, 2304, 3456, 3840, 5184, 6144, 9216, 5120

Offset: 1

Labos Elemer

nonn

			If n=20, then the number of divisors of C(20,k) is one of {1,6,8,16,24,40,48,64,80}, so a(20) = 80.

Michael De Vlieger, Table of n, a(n) for n = 1..1000

Cf. A000005, A001405, A034974.

Table[Max@ Map[DivisorSigma[0, #] &, Binomial[n, Range[0, n]]], {n, 53}] (* Michael De Vlieger, Mar 05 2017 *)

a(n) = vecmax(vector(n+1, k, numdiv(binomial(n,k-1)))); \\ Michel Marcus, Mar 05 2017

A048484 a(n) = abs(floor(n/2) - A048299(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 0, 1, 1, 1, 0, 0, 2, 3, 0, 0, 3, 3, 2, 2, 3, 3, 2, 7, 7, 1, 4, 4, 6, 5, 4, 4, 1, 2, 2, 2, 1, 1, 0, 0, 3, 3, 2, 6, 10, 7, 5, 4, 10, 5, 9, 8, 6, 3, 8, 7, 8, 8, 2, 1, 10, 10, 0, 0, 5, 4, 2, 2, 3, 7, 8, 7, 5, 5, 6, 3, 7, 7, 8, 4, 5, 6, 6, 11, 11, 10, 10, 4, 9, 8, 8, 7, 7, 6, 6, 5, 7, 13, 15

Offset: 1

Labos Elemer

nonn

			If n = 100 then the number of distinct primes at central C(100, 50) coefficient is 15, while the maximal is 18 which appears first at k = 35. Thus a(100) = 50 - 35 = 15.

Cf. A001221, A001405, A034974, A048275.

Table[Abs@ Floor[n/2] - Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &], {n, 100}] (* Michael De Vlieger, Aug 01 2017 *)

A048485 a(n) = floor(n/2) - A048475(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 3, 3, 3, 0, 3, 1, 1, 0, 1, 1, 0, 0, 4, 3, 3, 3, 2, 1, 0, 0, 0, 0, 2, 2, 1, 1, 3, 0, 0, 0, 1, 1, 5, 3, 4, 4, 3, 5, 0, 0, 8, 1, 7, 8, 2, 11, 1, 4, 4, 10, 3, 6, 11, 7, 7, 6, 6, 5, 4, 4, 7, 3, 2, 2, 5, 1, 0, 0, 0, 3, 2, 2, 1, 1, 0, 0, 0, 1, 0, 0, 3, 3

Offset: 1

Labos Elemer

			If n=51 then the number of divisors of the central binomial coefficient binomial(51,25) is 4608, while the maximal number of divisors of binomial(51,k) is 6144, which appears first at k=24; thus the deviation a(51) = |25-24| = 1 is small.

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

Cf. A000005, A001405, A034974, A048275.

a[n_] := Block[{d = DivisorSigma[0, Binomial[n, Range[0, n/2]]]}, Floor[n/2] - Position[d, Max[d], 1, 1][[1, 1]] + 1]; Array[a, 100] (* Giovanni Resta, May 14 2018 *)

A048621 a(n) = A001222(A001405(n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 11, 10, 11, 8, 9, 10, 11, 10, 11, 11, 12, 11, 12, 11, 12, 11, 12, 14, 15, 12, 13, 13, 14, 14, 15, 13, 14, 13, 14, 15, 16, 14, 15, 15, 16, 14, 15, 15, 16, 15, 16, 16, 17, 14, 15, 15, 16, 17, 18, 18, 19

Offset: 1

Labos Elemer

nonn

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

Cf. A001222, A001221, A046660, A034973, A034974.

Table[PrimeOmega[Binomial[n, Floor[n/2]]], {n, 1, 50}] (* G. C. Greubel, May 12 2017 *)

for(n=1,100, print1(bigomega(binomial(n,floor(n/2))), ", ")) \\ G. C. Greubel, May 12 2017

a(n) = A001222(A001405(n)).

Simpler name using formula by Joerg Arndt, May 14 2017

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

A036541 Deficit of central binomial coefficients in terms of number of prime factors: a(n) shows how many fewer prime factors the n-th central binomial coefficient has than n!.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 4, 4, 3, 3, 5, 5, 6, 6, 6, 5, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 8

Offset: 1

Labos Elemer

nonn

Primes not exceeding n/2 are missing from this kit of prime divisors. Note differences of consecutive deficits change sign like: 0,1,0,-2,0,-1,0,+2,0.

a(2n) = a(2n-1) unless n = 2^k for some k >= 1, in which case a(2n) = a(2n-1)-1. - Robert Israel, May 31 2016

			a(1000) = PrimePi(1000) - omega(binomial(1000, 500)) = 168 - 116 = 52.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001405, A000720, A034973, A034974.

N:= 1000: # to get a(1) .. a(N) G:= proc(p,n) local m,Ln,Lm;
   m:= floor(n/2);
   Ln:= convert(n,base,p);
   Lm:= convert(m,base,p);
   hastype(Ln[1..nops(Lm)]-Lm,negative)
end proc:
S[1]:= {}:
S[2]:= {}:
for n from 3 to N do
  if n::even then
     if n = 2^ilog2(n) then S[n]:= S[n-1] minus {2}
     else S[n]:= S[n-1]
     fi
  else
     S[n]:= (S[n-1] minus select(G,numtheory:-factorset(n),n)) union remove(G,numtheory:-factorset((n+1)/2),n);
  fi;
od:
seq(nops(S[i]),i=1..N); # Robert Israel, May 31 2016

Table[PrimePi@ n - PrimeNu[Binomial[n, Floor[n/2]]], {n, 105}] (* Michael De Vlieger, Jun 01 2016 *)

a(n) = omega(n!) - omega(binomial(n, floor(n/2))) = PrimePi(n) - omega(binomial(n, floor(n/2))).

cp's OEIS Frontend

A034973 Number of distinct prime factors in central binomial coefficients C(n, floor(n/2)), the terms of A001405.

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A056059 GCD of largest square and squarefree part of central binomial coefficients.

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A056057 The largest square which divides n-th central binomial coefficient.

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A056060 The powerfree part of the central binomial coefficients.

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A048275 a(n) = maximal value for number of divisors of C(n,k) for k=0..n.

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A048484 a(n) = abs(floor(n/2) - A048299(n)).

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A048485 a(n) = floor(n/2) - A048475(n).

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A048621 a(n) = A001222(A001405(n)).

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