A050608 -id:A050608 - cp's OEIS frontend

A046097 Values of n for which binomial(2n-1, n) is squarefree.

1, 2, 3, 4, 6, 9, 10, 12, 36

Offset: 1

No more terms up to 2^300. The sequence is finite by results of Sander and of Granville and Ramaré (see links). - Robert Israel, Dec 10 2015

Eric Weisstein's World of Mathematics, Binomial Coefficient.
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107.
J. W. Sander, Prime power divisors of binomial coefficients, Journal für die reine und angewandte Mathematik 430 (1992), 1-20.

Cf. A001700.

For a term to be here, it needs to be at least in the intersection of A048645, A051382, A050607, A050608 and an infinitude of similar sequences. The corresponding location in next-to-center column should be nonzero in A034931 (Pascal's triangle mod 4) and all similarly constructed fractal triangles (Pascal's triangle mod p^2).

[n: n in [1..150] | IsSquarefree(Binomial(2*n-1,n))]; // Vincenzo Librandi, Dec 10 2015

select(n -> numtheory:-issqrfree(binomial(2*n-1,n)), [$1..2000]); # Robert Israel, Dec 09 2015
N:= 300: # to find all terms <= 2^N
carries:= proc(n,m,p)
# number of carries when adding n + m in base p.
local A,B,C,j,nc, t;
   A:= convert(m,base,p);
   B:= convert(n,base,p);
C:= 0; nc:= 0;
   if nops(A) < nops(B) then A = [op(A),0$(nops(B)-nops(A))]
   elif nops(A) > nops(B) then B:= [op(B), 0$(nops(A)-nops(B))]
   fi;
for j from 1 to nops(A) do
    t:= C + A[j] + B[j];
    if t >= p then
       nc:= nc+1;
       C:= 1;
    else
       C:= 0
    fi
od:
nc;
end proc:
Cands:=  {seq(2^j,j=0..N), seq(seq(2^j + 2^k, k=0..j-1),j=1..N-1)}:
for i from 2 to 10 do
  Cands:= select(n -> carries(n-1,n,ithprime(i)) <= 1, Cands)
od:
select(n -> numtheory:-issqrfree(binomial(2*n-1,n)),Cands); # Robert Israel, Dec 10 2015

Select[ Range[1500], SquareFreeQ[ Binomial[ 2#-1, #]] &] (* Jean-François Alcover, Oct 25 2012 *)

is(n)=issquarefree(binomial(2*n-1,n)) \\ Anders Hellström, Dec 09 2015

James Sellers reports no further terms below 1500.
Michael Somos checked to 99999. Probably there are no more terms.
Mauro Fiorentini checked up to 2^64, as for n = 545259520, the binomial coefficient is a multiple of 5^4 and other possible exceptions have been checked (see Weisstein page for details).

A037461 a(n)=Sum{d(i)7^i: i=0,1,...,m}, where Sum{d(i)4^i: i=0,1,...,m} is the base 4 representation of n.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 10, 14, 15, 16, 17, 21, 22, 23, 24, 49, 50, 51, 52, 56, 57, 58, 59, 63, 64, 65, 66, 70, 71, 72, 73, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 114, 115, 119, 120, 121, 122, 147, 148, 149, 150, 154, 155, 156, 157

Offset: 1

Clark Kimberling

A number k is a term of this sequence if and only if 7 divides neither C(2*k-1,k) nor C(2*k,k).

			39 = 3*1 + 1*4 + 2*4^2 -> 3*1 + 1*7 + 3*7^2 = 108, so a(39) = 108. - _Clark Kimberling_, Jul 30 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A050608.

Table[FromDigits[RealDigits[n,4],7],{n,1,100}]
(* Clark Kimberling, Aug 02 2012 *)

A050607 Numbers k such that base 5 expansion matches (0|1|2)((0|1)(3|4))?(0|1|2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 75, 76, 77, 80, 81, 82, 85, 86, 87, 100, 101, 102, 105, 106, 107, 110, 111, 112, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 140, 141, 142, 145

Offset: 1

Antti Karttunen, Oct 24 1999

25 does not divide C(2s-1,s) = A001700(s) (nor C(2s,s) = A000984(s), central column of Pascal's triangle) if and only if s is one of the terms in this sequence.

Index entries for 5-automatic sequences.

Cf. A037453, A046097, A050608, A051382.

sub conv_x_base_n { my($x,$b) = @_; my ($r,$z) = (0,'');
do { $r = $x % $b; $x = ($x - $r)/$b; $z = "$r" . $z; } while(0 != $x);
return($z); }
for($i=0; $i <= 201; $i++) { if(("0" . conv_x_base_n($i,5)) =~ /^(0|1|2)*((0|1)(3|4))?(0|1|2)*$/) { print $i, ","; } }

a(1)=0 inserted by Georg Fischer, Jun 26 2021

A093700 Number of 9's immediately following the decimal point in the expansion of (3+sqrt(8))^n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 55, 56, 57

Offset: 1

Marvin Ray Burns, Apr 10 2004

Number of 0's immediately following the decimal point in the expansion of (3-sqrt(8))^n.

			Let n=10, (3+sqrt(8))^10= 45239073.9999999778... (the fractional part starts with seven 9's), so the 10th element in this sequence is 7.
The 132nd element is 100. The 1000th element is 765. The 1307th element is 1000.
The arrangement of repeating elements are like A074184 (Index of the smallest power of n >= n!) and A076539 (Numerators a(n) of fractions slowly converging to pi) and A080686 (Number of 19-smooth numbers <= n).

Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Experiment. Math. Volume 9, Issue 1 (2000), 3-12, Project Euclid - Cornell Univ (see Proposition 1).
Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Math Forum, Triangular Numbers That are Perfect Squares
Math Pages, On m = sqrt(sqrt(n) + sqrt(kn+1)) [Wrong link]
Robert Simms, Using counting numbers to generate Pythagorean triples

Cf. A003499, A050608, A080686, A076539, A074184.

For[n = 1, n < 999, n++, Block[{$MaxExtraPrecision = 50*n}, Print[ -Floor[Log[10, 1 - N[FractionalPart[(3 + 2Sqrt[2])^n], n]]] - 1]]]
f[n_] := Block[{}, -MantissaExponent[(3 - Sqrt[8])^n][[2]]]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Apr 10 2004 *)

Roughly, floor(3*n/4)

cp's OEIS Frontend

A046097 Values of n for which binomial(2n-1, n) is squarefree.

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A037461 a(n)=Sum{d(i)*7^i: i=0,1,...,m}, where Sum{d(i)*4^i: i=0,1,...,m} is the base 4 representation of n.

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A050607 Numbers k such that base 5 expansion matches (0|1|2)*((0|1)(3|4))?(0|1|2)*.

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A093700 Number of 9's immediately following the decimal point in the expansion of (3+sqrt(8))^n.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A037461 a(n)=Sum{d(i)7^i: i=0,1,...,m}, where Sum{d(i)4^i: i=0,1,...,m} is the base 4 representation of n.

A050607 Numbers k such that base 5 expansion matches (0|1|2)((0|1)(3|4))?(0|1|2).