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Numbers n that are not prime powers and such that gcd(binomial(2*n-1,n), n) = 1.
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%I #21 Sep 08 2022 08:46:15

%S 39,55,93,111,119,155,161,185,253,275,279,305,327,333,351,363,377,403,

%T 407,413,497,511,517,533,559,629,635,649,655,685,689,697,707,741,749,

%U 755,779,785,791,813,817,849,871,893,901,905,923,981,1003,1011,1027,1043

%N Numbers n that are not prime powers and such that gcd(binomial(2*n-1,n), n) = 1.

%C It seems there is a typo in the Gua and Zeng link, it gives 175 instead of 185 as a term.

%H Chai Wah Wu, <a href="/A268083/b268083.txt">Table of n, a(n) for n = 1..10000</a>

%H Victor J.W. Guo and Jiang Zeng, <a href="http://dx.doi.org/10.1016/j.jnt.2009.07.005">Factors of binomial sums from the Catalan triangle</a>, Journal of Number Theory 130 (2010) 172-186.

%t Select[Range[2,1100],!PrimePowerQ[#]&&GCD[Binomial[2#-1,#],#]==1&] (* _Harvey P. Dale_, May 26 2020 *)

%o (PARI) isok(n) = (n != 1) && !isprimepower(n) && (gcd(binomial(2*n-1,n), n) == 1);

%o (Magma) [n : n in [2..2000] | not IsPrimePower(n) and Gcd(Binomial(2*n-1,n), n) eq 1]; // _Vincenzo Librandi_, Jan 26 2016

%o (Python)

%o from __future__ import division

%o from fractions import gcd

%o from sympy import factorint

%o A268083_list, b = [], 1

%o for n in range(1,10**4):

%o if len(factorint(n)) > 1 and gcd(b,n) == 1:

%o A268083_list.append(n)

%o b = b*2*(2*n+1)//(n+1) # _Chai Wah Wu_, Jan 26 2016

%Y Cf. A000961, A088218, A268082.

%K nonn

%O 1,1

%A _Michel Marcus_, Jan 26 2016