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A268083 Numbers k that are not prime powers and such that gcd(binomial(2*k-1,k), k) = 1.

%I A268083 #29 Jun 29 2025 10:58:50
%S A268083 39,55,93,111,119,155,161,185,253,275,279,305,327,333,351,363,377,403,
%T A268083 407,413,497,511,517,533,559,629,635,649,655,685,689,697,707,741,749,
%U A268083 755,779,785,791,813,817,849,871,893,901,905,923,981,1003,1011,1027,1043
%N A268083 Numbers k that are not prime powers and such that gcd(binomial(2*k-1,k), k) = 1.
%C A268083 It seems there is a typo in the Gua and Zeng link, it gives 175 instead of 185 as a term.
%H A268083 Chai Wah Wu, <a href="/A268083/b268083.txt">Table of n, a(n) for n = 1..10000</a>
%H A268083 Victor J.W. Guo and Jiang Zeng, <a href="https://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 A268083 Select[Range[2,1100],!PrimePowerQ[#]&&GCD[Binomial[2#-1,#],#]==1&] (* _Harvey P. Dale_, May 26 2020 *)
%o A268083 (PARI) isok(n) = (n != 1) && !isprimepower(n) && (gcd(binomial(2*n-1,n), n) == 1);
%o A268083 (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 A268083 (Python)
%o A268083 from math import gcd
%o A268083 from sympy import factorint
%o A268083 A268083_list, b = [], 1
%o A268083 for n in range(1,10**4):
%o A268083     if len(factorint(n)) > 1 and gcd(b,n) == 1:
%o A268083         A268083_list.append(n)
%o A268083     b = b*2*(2*n+1)//(n+1) # _Chai Wah Wu_, Jan 26 2016
%Y A268083 Cf. A000961, A088218, A268082.
%K A268083 nonn
%O A268083 1,1
%A A268083 _Michel Marcus_, Jan 26 2016