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A268082 Numbers n such that gcd(binomial(2*n-1,n), n) is equal to 1. 3
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 93, 97, 101, 103, 107, 109, 111, 113, 119, 121, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Or numbers n such that A088218(n) is coprime to n.
The power of primes (A000961) are terms of this sequence.
From Robert Israel, Jan 26 2016: (Start)
By Lucas's theorem, these are the n such that for every prime p dividing n, no base-p digit of n is greater than the corresponding base-p digit of 2n-1. Equivalently (Kummer's theorem), there are no carries in base-p addition of n and n-1. Thus if p is odd, each base-p digit of n is less than p/2.
The only even terms are powers of 2.
All terms divisible by 3 are in A005836, and all terms divisible by 5 are in A037453. (End)
A082916 (after 0) lists the odd terms of this sequence. - Bruno Berselli, Jan 26 2015
LINKS
Victor J.W. Guo and Jiang Zeng, Factors of binomial sums from the Catalan triangle, Journal of Number Theory 130 (2010) 172-186.
Wikipedia, Kummer's theorem
Wikipedia, Lucas's theorem
EXAMPLE
For n=3, binomial(2*n-1, n) = binomial(5, 3) = 10 and 10 is coprime to 3, so 3 is in the sequence.
MAPLE
filter:= proc(n) local F, p;
if n::even then evalb(n = 2^padic:-ordp(n, 2))
else
F:= numtheory:-factorset(n);
for p in F do
if max(convert(n, base, p)) > p/2 then return false fi;
od;
true
fi
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 26 2016
MATHEMATICA
Select[Range@ 180, GCD[Binomial[2 # - 1, #], #] == 1 &] (* Michael De Vlieger, Jan 26 2016 *)
PROG
(PARI) isok(n) = gcd(binomial(2*n-1, n), n) == 1;
(PARI) lista(nn) = for(n=1, nn, if(gcd(binomial(2*n-1, n), n) == 1, print1(n, ", "))); \\ Altug Alkan, Jan 26 2016
(Magma) [n: n in [1..200] | Gcd(Binomial(2*n-1, n), n) eq 1]; // Vincenzo Librandi, Jan 26 2016
CROSSREFS
Sequence in context: A323306 A325247 A306013 * A247199 A087441 A326645
KEYWORD
nonn
AUTHOR
Michel Marcus, Jan 26 2016
STATUS
approved

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