A067434 -id:A067434 - cp's OEIS frontend

A383214 a(n) = A067434(n) - A383213(n).

1, 1, 0, 1, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 1, -2, 1, -1, 1, 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1

Offset: 1

Clark Kimberling, Apr 19 2025

sign

Least n such that a(n) = -3 is 7055; least n such than a(n) = 3 is 740.

Cf. A000984, A067634, A383213.

u = Table[PrimeNu[Binomial[2 n, n]], {n,  200}]     (* A067434 *)
v = Table[PrimeNu[Binomial[2 n, n + 1]], {n,  200}] (* A383213 *)
u - v

A071855 Numbers n such that A067434(n) = A000720(n) + A000005(n).

Original entry on oeis.org

1, 37, 41, 43, 47, 49, 58, 65, 69, 81, 85, 92, 128, 130, 135, 136, 154, 184, 208, 220, 228, 450, 540, 840

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

No further terms below 143260.

No further terms below 2*10^4. - Robert G. Wilson v

Select[ Range[10^3], Length[ FactorInteger[ Binomial[2#, # ]]] == PrimePi[ # ] + DivisorSigma[0, # ] & ]

for(n=1,1000,if(omega(binomial(2*n,n))=sum(i=1,n,isprime(i))+numdiv(n),print1(n,",")))

A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

Original entry on oeis.org

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

Offset: 1

Cino Hilliard, Jul 22 2005

a(n) < log(4)*n/log(n) < 7*n/(5*log(n)) for n > 1. - Reinhard Zumkeller, Mar 04 2008
Bertrand's postulate is equivalent to the formula a(n) >= 1 for all positive integers n. - Jonathan Vos Post, Jul 30 2008
Number of distinct prime factors > n of binomial(2*n,n). - T. D. Noe, Aug 18 2011
f(2, 2n) - f(3, n) < a(n) < f(3, 2n) - f(2, n) for n > 5889 where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). The constant 3 can be improved. - Charles R Greathouse IV, May 02 2012
For n >= 2, a(n) is the number of primes appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

F. Irschebeck, Einladung zur Zahlentheorie, BI Wissenschaftsverlag 1992, p. 40.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 181-182.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tsutomu Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.

Cf. A000720, A060715.
Cf. A067434 (number of prime factors in binomial(2*n,n)), A193990, A074990.
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

A108954 := proc(n)
    numtheory[pi](2*n)-numtheory[pi](n) ;
end proc: # R. J. Mathar, Nov 03 2017

Table[Length[Select[Transpose[FactorInteger[Binomial[2 n, n]]][[1]], # > n &]], {n, 100}] (* T. D. Noe, Aug 18 2011 *)
f[n_] := Length@ Select[ Range[n + 1, 2n], PrimeQ]; Array[f, 100] (* Robert G. Wilson v, Mar 20 2012 *)
Table[PrimePi[2n]-PrimePi[n],{n,90}] (* Harvey P. Dale, Mar 11 2013 *)

g(n) = for(x=1,n,y=primepi(2*x)-primepi(x);print1(y","))

from sympy import primepi
def A108954(n): return primepi(n<<1)-primepi(n) # Chai Wah Wu, Aug 19 2024

a(n) = A000720(2*n)-A000720(n).
For n > 1, a(n) = A060715(n). - David Wasserman, Nov 04 2005
Conjecture: G.f.: Sum_{i>0} Sum_{j>=i|i+j is prime} x^j. - Benedict W. J. Irwin, Mar 31 2017
From Wesley Ivan Hurt, Sep 20 2021: (Start)
a(n) = Sum_{k=1..n} A010051(2*n-k+1).
a(n) = Sum_{k=n*(n+1)/2+2..(n+1)*(n+2)/2} A010051(A128076(k)). (End)

A060308 Largest prime <= 2n.

Original entry on oeis.org

2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113, 113, 113, 113, 127, 127, 131

Offset: 1

Labos Elemer, Mar 27 2001

a(n) is the smallest k such that C(2n,n) divides k!. - Benoit Cloitre, May 30 2002
a(n) is largest prime factor of C(2n,n) = (2n)!/(n!)^2. - Alexander Adamchuk, Jul 11 2006
a(n) is also the largest prime in the interval [n,2n]. - Peter Luschny, Mar 04 2011
Odd prime p repeats (q-p)/2 times, where q > p is the next prime. In particular, every lesser of twin primes (A001359) occurs 1 time,  every lesser more than 3 of cousin primes (A023200) occurs 2 times, etc. - Vladimir Shevelev, Mar 12 2012

			n=1, 2n=2, p(1) = 2 = a(1) is the largest prime not exceeding 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Apart from initial term, same as A060265.
Cf. A007917 (largest prime <= n), A005843 (2n).
Cf. A020482, A049653, A060266, A060267, A060268, A060264.

a060308 = a007917 . a005843 -- Reinhard Zumkeller, May 25 2013

[NthPrime(#PrimesUpTo(2*n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

seq (prevprime(2*i+1), i=1..256);
seq(max(op(select(isprime,[$n..2*n]))),n=1..66); # Peter Luschny, Mar 04 2011

Table[Max[FactorInteger[(2n)!/(n!)^2]],{n,1,100}] (* Alexander Adamchuk, Jul 11 2006 *)
NextPrime[2*Range[80]+1,-1] (* Harvey P. Dale, Apr 23 2017 *)

a(n)=precprime(2*n) \\ Charles R Greathouse IV, May 24 2013

a(n) = Max[FactorInteger[(2n)!/(n!)^2]]. - Alexander Adamchuk, Jul 11 2006
a(n) = A006530(A000142(2*n)) and a(n) = A006530(A056040(2*n)). - Peter Luschny, Mar 04 2011
a(n) ~ 2*n as n tends to infinity. - Vladimir Shevelev, Mar 12 2012
a(n) = A007917(A005843(n)) = A226078(n, A067434(n)). - Reinhard Zumkeller, May 25 2013

More terms from Alexander Adamchuk, Jul 11 2006

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

Original entry on oeis.org

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

A081387 Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 4, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 4, 2, 3, 3, 2, 2, 2, 4, 3, 3, 4, 3, 4, 5, 4, 2, 2, 2, 3, 5, 5, 5, 5, 3, 2, 3, 2, 3, 3, 3

Offset: 1

Labos Elemer, Mar 27 2003

nonn

			For n = 14: binomial(28,14) = 40116600 = 2*2*2*3*3*3*5*5*17*19*23; unitary prime divisors: {17,19,23}; non-unitary prime divisors: {2,3,5}, so a(14) = 3.

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

Cf. A000984, A034444, A048105, A056169, A056170, A067434, A081387, A081388, A081389.

Table[Count[Transpose[FactorInteger[Binomial[2n,n]]][[2]],?(#>1&)],{n,110}] (* _Harvey P. Dale, Oct 08 2012 *)

a(n) = {my(e = factor(binomial(2*n, n))[, 2]); sum(k = 1, #e, e[k] > 1);} \\ Amiram Eldar, Oct 05 2024

a(n) = A056170(A000984(n)) = A001221(A000984(n)) - A081386(n) = A067434(n) - A081386(n).

A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 5, 7, 4, 9, 7, 4, 3, 7, 11, 8, 3, 11, 13, 2, 9, 5, 11, 13, 4, 5, 11, 13, 17, 4, 11, 13, 17, 19, 8, 3, 7, 13, 17, 19, 4, 7, 13, 17, 19, 23, 8, 25, 7, 17, 19, 23, 8, 27, 25, 17, 19, 23, 16, 9, 5, 17, 19, 23, 29, 2, 9, 5, 17, 19, 23, 29, 31

Offset: 0

Reinhard Zumkeller, May 25 2013

			.   n        initial rows               A000984(n)   A226047(n)
.  ---+------------------------------+-------------+------------
.   0   [1]                                      1
.   1   [2]                                      2            2
.   2   [2,3]                                    6            3
.   3   [4,5]                                   20            5
.   4   [2,5,7]                                 70            7
.   5   [4,9,7]                                252            9
.   6   [4,3,7,11]                             924           11
.   7   [8,3,11,13]                           3432           13
.   8   [2,9,5,11,13]                        12870           13
.   9   [4,5,11,13,17]                       48620           17
.  10   [4,11,13,17,19]                     184756           19
.  11   [8,3,7,13,17,19]                    705432           19
.  12   [4,7,13,17,19,23]                  2704156           23
.  13   [8,25,7,17,19,23]                 10400600           25
.  14   [8,27,25,17,19,23]                40116600           27
.  15   [16,9,5,17,19,23,29]             155117520           29
.  16   [2,9,5,17,19,23,29,31]           601080390           31
.  17   [4,27,5,11,19,23,29,31]         2333606220           31
.  18   [4,3,25,7,11,19,23,29,31]       9075135300           31
.  19   [8,3,25,7,11,23,29,31,37]      35345263800           37
.  20   [4,9,5,7,11,13,23,29,31,37]   137846528820           37 .

Reinhard Zumkeller, Rows n = 0..250 of triangle, flattened

Cf. A067434 (row lengths), A001316 (left edge), A060308 (right edge), A226047 (row maxima), A226083 (row minima), A000984 (row products).

Cf. A267823.

a226078 n k = a226078_tabf !! n !! k
a226078_row n = a226078_tabf !! n
a226078_tabf = map a141809_row a000984_list

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
b:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      seq(`if`(coeff(p, x, i)>0,
             i^coeff(p, x, i), NULL), i=1..degree(p))
    end:
T:= n-> `if`(n=0, 1, b(n)):
seq(T(n), n=0..30);  # Alois P. Heinz, May 25 2013

Table[Power @@@ FactorInteger[(2n)!/n!^2] , {n, 0, 30}] // Flatten (* Jean-François Alcover, Jul 29 2015 *)

row(n)= if(n<1, [1], [ e[1]^e[2] |e<-Col(factor(binomial(2*n, n)))]); \\ Ruud H.G. van Tol, Nov 18 2024

T(n,k) = A141809(A000984(n),k) for k = 0..A067434(n)-1.

A081386 Number of unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of those prime factors in C(2n,n), whose exponent equals one.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 3, 4, 4, 4, 5, 5, 4, 3, 5, 7, 6, 7, 7, 8, 9, 9, 6, 7, 7, 7, 8, 11, 12, 11, 11, 11, 12, 12, 12, 13, 13, 13, 11, 13, 12, 14, 13, 13, 15, 14, 14, 14, 15, 16, 16, 16, 17, 19, 18, 17, 18, 19, 18, 19, 18, 18, 18, 20, 18, 21, 22, 20, 20, 20, 20, 20, 20, 19, 21, 21, 24, 23

Offset: 1

Labos Elemer, Mar 27 2003

nonn

			n=10: C(20,10) = 184756 = 2*2*11*13*17*19; unitary-p-divisors = {11,13,17,19}, so a(10)=4.

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A000984, A067434, A081387-A081389, A034444, A048105, A056170, A056169.

Table[Function[m, Count[Divisors@ m, k_ /; And[PrimeQ@ k, GCD[k, m/k] == 1]]]@ Binomial[2 n, n], {n, 50}] (* Michael De Vlieger, Dec 17 2016 *)

a(n) = my(f=factor(binomial(2*n, n))); sum(k=1, #f~, f[k,2] == 1); \\ Michel Marcus, Dec 18 2016

a(n) = A056169(A000984(n)).

A226047 Largest prime power dividing binomial(2n, n).

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 13, 17, 19, 19, 23, 25, 27, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 49, 49, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 81, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113

Offset: 1

Charles R Greathouse IV, May 24 2013

The first indices n for which a(n) < a(n-1) are 123, 315, 366, 671, 1095, 1098, 1204, 1565, 6095, 7326, 9843, 39065, 58828, 88575, 88578, 195315, 195320, 265722, 265725 and 709937. - Giovanni Resta, May 24 2013

a(n) = maximum of n-th row in A226078. - Reinhard Zumkeller, May 25 2013

			Binomial(10, 5) = 2^2 * 3^2 * 7 and so a(5) = max({2^2, 3^2, 7}) = 3^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged 5 (1932), pp. 194-198.

Cf. A000961, A000984, A060308, A067434, A226083.

a226047 = maximum . a226078_row  -- Reinhard Zumkeller, May 25 2013

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
a:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      max(seq(i^coeff(p, x, i), i=1..degree(p)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 24 2013

cnt[n_, p_] := (n - Total@IntegerDigits[n, p])/(p-1); a[n_] := Block[{k = 2*n, p, e}, While[! PrimePowerQ[k] || ({p, e} = FactorInteger[k][[1]]; cnt[2*n , p] - 2 cnt[n, p] != e), k--]; k]; Array[a, 60] (* Giovanni Resta, May 24 2013 *)
Table[Max[Select[Divisors[Binomial[2 n,n]],PrimePowerQ]],{n,60}] (* Harvey P. Dale, Feb 26 2024 *)

ord(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(p=precprime(2*n));forstep(k=2*n,p+1,-1, my(q,e=isprimepower(k, &q)); if(e && e == ord(2*n,q)-2*ord(n,q), return(k)));p /* requires PARI v.2.5 or later */

A226047(n)={for(k=2,#n=factor(binomial(2*n,n))~,factorback(n[,k-1]~)>factorback(n[,k]~) && n[,k]=n[,k-1]);factorback(n[,#n]~)} \\ highly unoptimized, not suitable for n>>10^4. - M. F. Hasler, May 24 2013

Erdős proved that a(n) <= 2n.

A193990 Number of distinct prime factors <= n of binomial(2*n,n).

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Aug 17 2011

nonn

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

Cf. A067434 (number of prime factors in binomial(2*n,n)), A108954.

a := n -> nops(numtheory:-factorset(binomial(2*n,n)) minus select(isprime, {$n+1..2*n})): seq(a(n), n=1..82); # Peter Luschny, Oct 31 2015

Table[Length[Select[Transpose[FactorInteger[Binomial[2 n, n]]][[1]], # <= n &]], {n, 100}]

valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(s); forprime(p=2, n, if(valp(2*n, p)>2*valp(n, p), s++)); s \\ Charles R Greathouse IV, May 25 2013

cp's OEIS Frontend

A383214 a(n) = A067434(n) - A383213(n).

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A071855 Numbers n such that A067434(n) = A000720(n) + A000005(n).

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A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

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A060308 Largest prime <= 2n.

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

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A081387 Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.

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A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

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