A129508 -id:A129508 - cp's OEIS frontend

A030979 Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.

0, 1, 10, 756, 757, 3160, 3186, 3187, 3250, 7560, 7561, 7651, 20007, 59548377, 59548401, 45773612811, 45775397187, 237617431723407, 24991943420078301, 24991943420078302, 24991943420078307, 24991943715007536, 24991943715007537

Offset: 1

Shawn Godin (sgodin(AT)onlink.net)

nonn

By Lucas's theorem, binomial(2k,k) is not divisible by a prime p iff all base-p digits of k are smaller than p/2.
Ronald L. Graham offered $1000 to the first person who could settle the question of whether this sequence is finite or infinite. He remarked that heuristic arguments show that it should be infinite, but finite if it is required that binomial(2k,k) is prime to 3, 5, 7 and 11, with k = 3160 probably the last k which has this property.
The Erdős et al. paper shows that for any two odd primes p and q there are an infinite number of k for which gcd(p*q,binomial(2k,k))=1; i.e., p and q do not divide binomial(2k,k). The paper does not deal with the case of three primes. - T. D. Noe, Apr 18 2007
Pomerance gives a heuristic suggesting that there are on the order of x^0.02595... terms up to x. - Charles R Greathouse IV, Oct 09 2015

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.

Christopher E. Thompson, Table of n, a(n) for n = 1..1374 (complete up to 10^70, extends first 62 terms computed by Max Alekseyev).
Christian Ballot, Divisibility of the middle Lucasnomial coefficient, Fib. Q., 55 (2017), 297-308.
Ernie Croot, Hamed Mousavi and Maxie Schmidt, On a conjecture of Graham on the p-divisibility of central binomial coefficients, arXiv:2201.11274 [math.NT], 2022.
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, doi:10.2307/2005464.
Gennady Eremin, Factoring Middle Binomial Coefficients, arXiv:2003.01494 [math.CO], 2020.
R. D. Mauldin and S. M. Ulam, Mathematical problems and games, Adv. Appl. Math. 8 (3) (1987) 281-344.
C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.
Wikipedia, Lucas' theorem
Han Yu, Fractal projections with an application in number theory, arXiv:2004.05924 [math.NT], 2020.

Cf. A129488, A129489, A129508, A151750.

lim=10000; Intersection[Table[FromDigits[IntegerDigits[k,2],3], {k,0,lim}], Table[FromDigits[IntegerDigits[k,3],5], {k,0,lim}], Table[FromDigits[IntegerDigits[k,4],7], {k,0,lim}]] (* T. D. Noe, Apr 18 2007 *)

fval(n,p)=my(s);while(n\=p,s+=n);s
is(n)=fval(2*n,3)==2*fval(n,3) && fval(2*n,5)==2*fval(n,5) && fval(2*n,7)==2*fval(n,7) \\ Charles R Greathouse IV, Oct 09 2015

Intersection of A005836, A037453 and A037461. - T. D. Noe, Apr 18 2007

More terms from Naohiro Nomoto, May 06 2002
Additional comments from R. L. Graham, Apr 25 2007
Additional comments and terms up 3^41 in b-file from Max Alekseyev, Nov 23 2008
Additional terms up to 10^70 in b-file from Christopher E. Thompson, Nov 06 2015

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

cp's OEIS Frontend

A030979 Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.

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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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