A030979 -id:A030979 - cp's OEIS frontend

A129508 Numbers k such that 3 and 5 do not divide binomial(2*k, k).

0, 1, 10, 12, 27, 30, 31, 36, 37, 252, 255, 256, 280, 282, 756, 757, 760, 810, 811, 3160, 3162, 3186, 3187, 3250, 3252, 3276, 3277, 3280, 6561, 6562, 6885, 6886, 6912, 6925, 7536, 7537, 7560, 7561, 7626, 7627, 7650, 7651, 19686, 19687, 20007, 20010, 20011

Offset: 1

T. D. Noe, Apr 18 2007

nonn

The Erdos paper proves that for any two odd primes p and q, there are an infinite number of k for which gcd(p*q, binomial(2*k, k)) = 1; i.e., p and q do not divide binomial(2*k, k).

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Erdos, R. L. Graham, I. Z. Russa, and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.

Cf. A030979 (k such that 3, 5 and 7 do not divide binomial(2*k, k)).

lim=10000; Intersection[Table[FromDigits[IntegerDigits[k,2],3], {k,0,lim}], Table[FromDigits[IntegerDigits[k,3],5], {k,0,lim}]]

valp(n, p)=my(s); while(n\=p, s+=n); s
is(n)=valp(2*n, 3)==2*valp(n, 3) && valp(2*n, 5)==2*valp(n, 5) \\ Charles R Greathouse IV, Feb 03 2016

Intersection of A005836 and A037453.

A129488 Smallest odd prime dividing binomial(2n,n).

Original entry on oeis.org

3, 5, 5, 3, 3, 3, 3, 5, 11, 3, 7, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 5, 3, 7, 7, 3, 3, 3, 3, 7, 7, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 3, 5, 5, 3, 3, 3, 3, 5, 5, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 2

T. D. Noe, Apr 17 2007

nonn

The Erdős paper calls this function g(n) and states that it is not known whether it is bounded. Currently, g(3160)=13 is the greatest known value of g. See A129489.

T. D. Noe, Table of n, a(n) for n = 2..5000
P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.

Cf. A030979 (n such that g(n)>=11), A129489, A266366.

Table[Transpose[FactorInteger[Binomial[2n,n]]][[1,2]], {n,2,150}]

a(n)=my(k);forprime(p=3,default(primelimit),k=1;while((k*=p)<=2*n,if(n/k-n\k>1/2,return(p)))) \\ Charles R Greathouse IV, Dec 19 2011

A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

Original entry on oeis.org

3, 10, 10, 3160

Offset: 1

T. D. Noe, Apr 17 2007

The Erdős paper states that it is not known whether the smallest odd prime factor, called g(n), of binomial(2n,n) is bounded. See A129488 for the function g(n). Lucas' Theorem for binomial coefficients can be used to quickly determine whether a prime p divides binomial(2n,n) without computing the binomial coefficient. It is probably a coincidence that 3, 10 and 3160 are all triangular numbers.

Extensive calculations show that if a(5) exists, it is either an integer greater than 13^12 or if it is a triangular number then it is greater than 2^63. [Comment modified by Robert Israel, Jan 27 2016]

			For n=1, binomial(6,3)=20, which is not divisible by 3.
For n=2 and n=3, binomial(20,10)=184756 is not divisible by 3, 5 and 7.
For n=4, binomial(6320,3160), a 1901-digit number, is not divisible by 3, 5, 7 and 11.

P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
Eric Weisstein's World of Mathematics, Lucas Correspondence Theorem

Cf. A000984, A129488, A030979 (n such that g(n)>=11), A266366, A267823.

Table[k = 2; While[AnyTrue[Prime@ Range[2, n + 1], Divisible[Binomial[2 k, k], #] &], k++]; k, {n, 4}] (* Michael De Vlieger, Jan 27 2016, Version 10 *)

isok(kk, n) = {for (j=2, n+1, if (kk % prime(j) == 0, return (0));); return (1);}
a(n) = {my(k = 2); while (! isok(binomial(2*k,k), n), k++); k;} \\ Michel Marcus, Jan 11 2016

a(n) <= A266366(n+1) for n > 0. - Jonathan Sondow, Jan 27 2016

A196747 Numbers n such that 3 does not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 6, 7, 8, 18, 19, 20, 24, 25, 26, 54, 55, 56, 60, 61, 62, 72, 73, 74, 78, 79, 80, 162, 163, 164, 168, 169, 170, 180, 181, 182, 186, 187, 188, 216, 217, 218, 222, 223, 224, 234, 235, 236, 240, 241, 242, 486, 487, 488, 492, 493, 494, 504, 505, 506, 510

Offset: 1

Peter Luschny, Oct 06 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196748, A196749, A196750.

SwingExp := proc(m,n) local p, q; p := m;
do q := iquo(n,p);
   if (q mod 2) = 1 then RETURN(1) fi;
   if q = 0 then RETURN(0) fi;
   p := p * m;
od end:
Search := proc(n,L) local m, i, r; m := n;
for i in L do r := SwingExp(i,m);
   if r <> 0 then RETURN(NULL) fi
od; n end:
A196747_list := n -> Search(n,[3]):  # n is a search limit

(* A naive solution *) sf[n_] := n!/Quotient[n, 2]!^2; Select[Range[0, 600], ! Divisible[sf[#], 3] &] (* Jean-François Alcover, Jun 28 2013 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=my(t=valp(n,3)); t%2==0 && 2*valp(n\2,3)==t \\ Charles R Greathouse IV, Feb 02 2016

A196748 Numbers n such that 3 and 5 do not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 20, 24, 54, 60, 61, 62, 72, 73, 74, 504, 510, 511, 512, 560, 564, 1512, 1513, 1514, 1520, 1620, 1621, 1622, 6320, 6324, 6372, 6373, 6374, 6500, 6504, 6552, 6553, 6554, 6560, 13122, 13123, 13124, 13770, 13771, 13772, 13824, 13850, 15072, 15073, 15074

Offset: 1

Peter Luschny, Oct 06 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
P. Erdos, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196747, A196749, A196750.

# The function Search is defined in A196747.
A196748_list := n -> Search(n,[3,5]):  # n is a search limit

(* A naive solution *) sf[n_] := n!/Quotient[n, 2]!^2; Select[Range[0, 16000], !Divisible[sf[#], 3] && !Divisible[sf[#], 5] &] (* Jean-François Alcover, Jun 28 2013 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=valp(n,3)==2*valp(n\2,3) && valp(n,5)==2*valp(n\2,5) \\ Charles R Greathouse IV, Feb 02 2016

A196749 Numbers n such that 3, 5 and 7 do not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 20, 1512, 1513, 1514, 6320, 6372, 6373, 6374, 6500, 15120, 15121, 15122, 15302, 40014, 119096754, 119096802, 91547225622, 91550794374

Offset: 1

Peter Luschny, Oct 06 2011

P. Erdos, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196747, A196748, A196750.

# The function Search is defined in A196747.
A196749_list := n -> Search(n,[3,5,7]):  # n is a search limit

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=valp(n,3)==2*valp(n\2,3) && valp(n,5)==2*valp(n\2,5) && valp(n,7)==2*valp(n\2,7) \\ Charles R Greathouse IV, Feb 02 2016

A196750 Numbers n such that 3, 5, 7 and 11 do not divide swing(n) = A056040(n).

Original entry on oeis.org

0, 1, 2, 6320

Offset: 1

Peter Luschny, Oct 06 2011

It is conjectured that there are no other terms.

Peter Luschny, On the prime factors of the swinging factorial.

Cf. A005836, A129508, A030979, A151750, A196747, A196748, A196749.

# The function Search is defined in A196747.
A196750_list := n -> Search(n,[3,5,7,11]):  # n is a search limit

valp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=valp(n,3)==2*valp(n\2,3) && valp(n,5)==2*valp(n\2,5) && valp(n,7)==2*valp(n\2,7) && valp(n,11)==2*valp(n\2,11) \\ Charles R Greathouse IV, Feb 02 2016

A266366 Least k such that prime(n) is the smallest odd prime factor of C(2k,k).

Original entry on oeis.org

2, 3, 12, 10, 3160

Offset: 2

Jonathan Sondow, Jan 18 2016

If n>0, then a(n+2) >= A129489(n) = least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

It is not known whether any more terms exist. See A129489 for bounds, comments and references.

			C(2,1) = 2, C(4,2) = 6 = 2 * 3, C(6,3) = 20 = 2^2 * 5, and 3 = prime(2), 5 = prime(3), so a(2) = 2 and a(3) = 3.

Cf. A000984, A129488, A129489, A030979.

valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(q=prime(n),k=1); while(k++, forprime(p=3,q-1, if(valp(2*k, p)>2*valp(k, p), next(2))); if(valp(2*k, q)>2*valp(k, q), return(k))) \\ Charles R Greathouse IV, Feb 03 2016

A373469 Least odd k such that C(2k, k) == 1 (mod A007775(n)), or 0 if no such k exists.

Original entry on oeis.org

1, 17, 13, 2383, 37, 3, 3391, 185, 129, 419, 95, 139, 7, 7373, 497, 21, 89, 27, 319, 7, 23, 191, 277, 25, 33635, 137, 1957, 347, 879, 889, 47, 57, 411, 263, 63, 57, 63, 143, 62561, 363, 1679, 861, 285735, 1017, 545, 2605, 913, 1873, 735, 206349, 817, 407, 485, 49, 7605, 179817

Offset: 1

M. F. Hasler, Jul 12 2024

A007775 lists the odd numbers not divisible by 3 or 5. It seemed that these are exactly the odd numbers not in A086748 (= odd m such that C(2k,k) == 1 (mod m) has no odd solution k), i.e., the numbers in A086748 would exactly be the odd multiples of 3 and 5, but so far there was no proof or disproof for that. The present sequence gives an explicit proof, if it exists, for each x in A007775, that x is not in A086748.
It is highly possible that a(n) = 0 for n with m = A007775(n) divisible by three or more distinct primes, in which case values of k such that C(2k,k) coprime to m, let alone C(2k,k) == 1 (mod m), are very sparse and possibly finite. See A030979 for a similar problem. - Max Alekseyev, Jul 14 2024
Examples for moduli that have 3 distinct prime factors >5: a(603) = 57 associated with modulus A007775(603) = 2261 = 7*17*19.  a(4333) = 23 associated with modulus A007775(4333) = 16247 = 7*11*211. a(6621) = 1709 associated with 11*37*61. a(6797)=19999 assocated with 7*11*331. - R. J. Mathar, Aug 09 2024

M. F. Hasler and Max Alekseyev, Table of n, a(n) for n = 1..266

Cf. A007775 (odd numbers not divisible by 3 or 5), A086748 (odd m such that C(2k,k)==1 (mod m) has no odd solution k).

Cf. A030979.

/* helper function: compute C(n,k) mod prime p */
LucasT(n,k,p)={if(n>=k, my(kp = digits(k,p), np = digits(n,p)[-#kp..-1]); prod(i=1, #kp, binomial(np[i], kp[i]), Mod(1,p)))}
is1(k,f)={for(i=1,matsize(f)[1], LucasT(2*k, k, f[i,1])==1||return); vecmax(f[,2])==1 || binomial(2*k,k)%factorback(f)==1}
apply( {A373469(n, m=A007775(n), f=factor(m))=!f || forstep(k=3, oo, 2, is1(k,f) && return(k))}, [1..50])

a(43)-a(56) from Max Alekseyev, Jul 12 2024

A295562 List of numbers whose middle Fibonomial coefficient (2n,n)_F is prime to 105.

Original entry on oeis.org

1, 1312, 3256, 3257, 3936, 3937, 4000, 4001, 4032, 38880, 38881, 39000, 39001, 19928280, 19928281, 21975136, 21975137, 21975305, 21975312, 22054032

Offset: 1

N. J. A. Sloane, Nov 28 2017

From Charlie Neder, Mar 04 2019: (Start)
For the middle Fibonomial coefficient (2n,n)_F to be coprime to a prime p, we must have that the integral part of n/A001177(p) has its base-p digits all < p/2 and its fractional part is < 1/2.
Next term > 10^8. (End)

Christian Ballot, Divisibility of the middle Lucasnomial coefficient, Fib. Q., 55 (2017), 297-308.

Cf. A003267, A030979.

Cf. A037461, A037543, A005836.

Numbers k congruent to 0 or 1 modulo 8 such that floor(k/4) is in A005836, k is in A037543, and floor(k/8) is in A037461. - Charlie Neder, Mar 04 2019

a(10)-a(20) from Charlie Neder, Mar 04 2019

cp's OEIS Frontend

A129508 Numbers k such that 3 and 5 do not divide binomial(2*k, k).

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A129488 Smallest odd prime dividing binomial(2n,n).

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A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

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A196747 Numbers n such that 3 does not divide swing(n) = A056040(n).

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A196748 Numbers n such that 3 and 5 do not divide swing(n) = A056040(n).

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A196749 Numbers n such that 3, 5 and 7 do not divide swing(n) = A056040(n).

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A196750 Numbers n such that 3, 5, 7 and 11 do not divide swing(n) = A056040(n).

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Maple

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A266366 Least k such that prime(n) is the smallest odd prime factor of C(2k,k).

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