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%I #27 Oct 05 2020 13:16:48
%S 21,35,39,50,55,57,63,65,75,77,85,93,98,111,115,119,129,133,143,155,
%T 161,171,175,183,185,187,189,201,203,205,209,215,217,219,221,235,237,
%U 242,245,247,253,259,265,275,279,291,299,301,305,309,319,323,325,327,329,333,335,338,341
%N Numbers which are coprime to the sum of their divisors, but are neither primes nor perfect powers.
%C Intersection of A003624 and A106543. - _Michel Marcus_, Sep 30 2014
%C Duffinian numbers (A003624) which are not perfect powers (A001597). - _Robert G. Wilson v_, Oct 02 2014
%H Amiram Eldar, <a href="/A248020/b248020.txt">Table of n, a(n) for n = 1..10000</a>
%e 21 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.
%e 50 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 2, 5, 10, 25, and 50 sum to 93, which is coprime to 50.
%t perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ Range@ 350, !PrimeQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 && !perfectPowerQ[ #] &]
%t cpQ[n_]:=CoprimeQ[n,DivisorSigma[1,n]]&&!PrimeQ[n]&&GCD@@ FactorInteger[ n][[All,2]]<2; Select[Range[2,400],cpQ] (* _Harvey P. Dale_, Oct 05 2020 *)
%o (PARI) forcomposite(n=1, 1e3, if(gcd(n, sigma(n))==1, if(!ispower(n), print1(n, ", ")))) \\ _Felix Fröhlich_, Oct 25 2014
%K nonn,easy
%O 1,1
%A _Robert G. Wilson v_, Sep 29 2014
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