A034973 -id:A034973 - cp's OEIS frontend

A034974 Number of divisors of binomial(n, floor(n/2)), the terms of A001405.

1, 2, 2, 4, 4, 6, 4, 8, 12, 18, 16, 24, 24, 32, 24, 48, 32, 48, 32, 48, 96, 128, 64, 96, 144, 192, 288, 384, 384, 480, 192, 384, 512, 768, 768, 1152, 1152, 1536, 1536, 2304, 1536, 2048, 1536, 2048, 3072, 3840, 2304, 3456, 3456, 4608, 4608, 6144, 3072, 3840, 6144

Offset: 1

Labos Elemer

nonn

			a(59) = d(C(59,29)) = d(59132290782430712) = 8192 = 4*2^11.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001405, A034973.

Table[DivisorSigma[0,Binomial[n,Floor[n/2]]],{n,60}] (* Harvey P. Dale, Jun 12 2012 *)

a(n) = numdiv(binomial(n, n\2)); \\ Michel Marcus, Feb 25 2014

a(n) = A000005(A001405(n)). - Michel Marcus, Feb 25 2014

A048273 Maximal number of distinct prime factors in binomial coefficients C(n,k) for k = 0,...,n.

Original entry on oeis.org

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

Offset: 0

Labos Elemer

nonn

			If n = 51 and k runs from 0 to 51, then a maximum of 11 distinct prime factors arise, for k = 20, 21, 22, 23, 28, 29, 30, 31.

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A001405, A034973, A001221, A048571.

Table[Max[Table[b = Binomial[n, k]; If[b == 1, 0, Length[FactorInteger[b]]], {k, 0, n}]], {n, 0, 100}] (* T. D. Noe, Apr 03 2012 *)

A048633 Largest squarefree number dividing n-th central binomial coefficient C(n,[ n/2 ]).

Original entry on oeis.org

1, 2, 3, 6, 10, 10, 35, 70, 42, 42, 462, 462, 858, 858, 2145, 4290, 24310, 24310, 92378, 92378, 176358, 176358, 1352078, 1352078, 520030, 520030, 222870, 222870, 6463230, 6463230, 100180065, 200360130, 129644790, 129644790, 907513530

Offset: 1

Labos Elemer

nonn

a(2k+1)=a(2k+2) unless 2k+1 is in A000225, in which case a(2k+2)=2*a(2k+1). - Robert Israel, Jan 21 2020

			n=10: C(10,5)=252=2*2*3*3*7. The largest squarefree number dividing the 10th central binomial coefficient is 2*3*7=42. Thus a(10)=42

Robert Israel, Table of n, a(n) for n = 1..3364

Equals A007947(A001405(n)). Cf. A034973, A000225.

See A056058 for another version.

[&*PrimeDivisors(Binomial(n, Floor(n/2))): n in [1..35]]; // Marius A. Burtea, Jan 21 2020

f:= n -> convert(numtheory:-factorset(binomial(n,floor(n/2))),`*`):
map(f, [$1..50]); # Robert Israel, Jan 21 2020

Table[Last@ Select[Divisors@ Binomial[n, Floor[n/2]], SquareFreeQ], {n, 35}] (* Michael De Vlieger, Feb 05 2017 *)

a(n)=factorback(factor(binomial(n,n\2))[,1]) \\ Charles R Greathouse IV, Nov 05 2017

A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 2, 2, 3, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Jul 27 2000

nonn

Number of prime divisors of the largest square dividing A001405(n). (A prime divisor is nonunitary iff its exponent exceeds 1.)

			For n=10, binomial(10, 5) = 252 = 2*2*3*3*7 has 3 prime divisors of which only one, p=7, is unitary, while 2 and 3 are not. So a(10)=2.
For n=256, binomial(256, 128) also has only 2 prime divisors (3 and 13) whose exponents exceed 1 (4 and 2, respectively), thus a(256)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001405, A034444, A034973, A039593, A056057, A056173.

Table[Count[FactorInteger[Binomial[n, Floor[n/2]]][[All, -1]], e_ /; e > 1], {n, 105}] (* Michael De Vlieger, Mar 05 2017 *)

a(n)=omega(core(binomial(n, n\2), 1)[2]) \\ Charles R Greathouse IV, Mar 09 2017

a(n) = A001221(A000188(A001405(n))).

a(n) = A001221(A056057(n)).

Edited by Jon E. Schoenfield, Mar 05 2017

A056173 Number of unitary prime divisors of central binomial coefficient C(n, floor(n/2)) (A001405).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 3, 3, 3, 3, 4, 5, 4, 5, 4, 5, 5, 6, 5, 4, 4, 3, 3, 5, 5, 6, 7, 7, 6, 8, 7, 7, 7, 9, 8, 9, 9, 9, 9, 6, 6, 8, 7, 7, 7, 7, 7, 8, 8, 11, 11, 12, 12, 11, 11, 11, 11, 10, 11, 13, 12, 13, 12, 12, 12, 14, 13, 13, 13, 13, 13, 11, 11, 14, 13, 12, 12, 14, 14, 13, 13, 13

Offset: 1

Labos Elemer, Jul 27 2000

nonn

A prime divisor is unitary iff its exponent equals 1.

			For n = 10: binomial(10,5) = 252 = 2*2*3*3*7 has 3 prime factors of which only one, p = 7, is unitary. So a(10) = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001405, A034973, A056169, A056175.

Array[Function[k, Count[FactorInteger[k][[All, 1]], ?(CoprimeQ[#, k/#] &)]]@ Binomial[#, Floor[#/2]] &, 87]  (* _Michael De Vlieger, Oct 26 2017 *)

a(n) = vecsum(apply(x -> if(x > 1, 0, 1), factor(binomial(n, n\2))[,2])); \\ Amiram Eldar, Jul 22 2024

a(n) = A056169(A001405(n)). - Michel Marcus, Oct 27 2017 [corrected by Amiram Eldar, Jul 22 2024]

A039593 Number of unitary divisors of central binomial coefficients.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 256, 256, 256, 512, 512, 512, 512, 1024, 1024, 1024, 1024, 1024, 1024, 512, 512, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 4096, 4096

Offset: 1

Labos Elemer

nonn

As in A034444, all terms are powers of 2.

			At n=5, the central binomial coefficient is 10, having 4 divisors, each of which is unitary, so a(5)=4.

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

Cf. A001405, A034444, A034973.

a[n_] := 2^PrimeNu[Binomial[n, Floor[n/2]]]; Array[a, 56] (* Amiram Eldar, Oct 06 2019 *)

a(n) = A034444(A001405(n)) = 2^A034973(n).

A048486 Values of k for which the earliest maximal value of A001221(C(k,j)) is j = floor(k/2).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 12, 13, 16, 17, 40, 41, 64, 65, 107, 108, 132, 133, 219, 220, 288, 340, 341, 400, 401, 419, 420, 421, 556, 576, 608, 651, 660, 661, 804, 809, 810, 811, 936, 937, 1020, 1054, 1055, 1063, 1255, 1256, 1307, 1308, 1368, 1408, 1409, 1555, 1556

Offset: 1

Labos Elemer

nonn

k is in the sequence if omega(C(k,j)) is a maximum for j = floor(k/2) and not a maximum for j < floor(k/2).

			If n = 16 and k = 0, ..., 16 then r = 0,1,3,3,4,4,4,4,5,4,4,4,4,3,3,1,0. The maximum of A001221(C(16,k)) values, i.e. 5 is appears at k = 8, the center. Thus 16 is in this sequence.

Cf. A001221, A001405, A034973, A048273, A048571.

Select[Range@ 500, Function[n, Min@ MaximalBy[Range[0, n], PrimeNu@ Binomial[n, #] &] == Floor[n/2]]] (* Michael De Vlieger, Aug 01 2017 *)

More terms from Michael De Vlieger, Aug 01 2017

Title clarified by Sean A. Irvine, Jun 18 2021

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

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))).

A048196 Numbers k for which binomial(k, floor(k/2)) has the same number of unitary and non-unitary divisors.

Original entry on oeis.org

14, 22, 33, 42, 44, 56, 57, 59, 74, 107, 113, 115, 1568, 1571

Offset: 1

Labos Elemer

Next term > 10^8. - David A. Corneth, May 14 2018

Numbers k where b = binomial(k, floor(k/2)) is of the form p_i ^ e_i where p_i is the i-th prime in the factorization of b, e_i = 1 except exactly one e_i = 3 for i > 1. - David A. Corneth, May 13 2018

			At k=59, the corresponding binomial coefficient, binomial(59,29) has 8192 divisors, of which 4096 are unitary and 4096 are not.

David A. Corneth, PARI link

Cf. A001405, A034444, A034973, A048105, A048109.

isok(n) ={ n=binomial(n, floor(n/2)); sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); } \\ Joerg Arndt, May 13 2018

\\ much faster:
isok(n) ={ n=binomial(n, floor(n/2)); my(u=1<Joerg Arndt, May 13 2018

\\ for a still faster program see the Corneth link.

A034444(A001405(k)) = A048105(A001405(k)).

a(9) .. a(14) from Joerg Arndt, May 13 2018

cp's OEIS Frontend

A034974 Number of divisors of binomial(n, floor(n/2)), the terms of A001405.

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A048273 Maximal number of distinct prime factors in binomial coefficients C(n,k) for k = 0,...,n.

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A048633 Largest squarefree number dividing n-th central binomial coefficient C(n,[ n/2 ]).

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A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

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A056173 Number of unitary prime divisors of central binomial coefficient C(n, floor(n/2)) (A001405).

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A039593 Number of unitary divisors of central binomial coefficients.

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A048486 Values of k for which the earliest maximal value of A001221(C(k,j)) is j = floor(k/2).

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