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%I #21 Sep 08 2022 08:46:26
%S 2,3,4,5,6,16,17,256,257,65536,65537,4294967296
%N Numbers k such that phi(binomial(k,2)) is a power of 2.
%C Every Fermat prime appears in this sequence.
%C A number greater than 2^32 is in this sequence if and only if it is a Fermat prime.
%D M. Krizek, F. Luca and L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, p. 86.
%D F. Luca, Pascal's triangle and constructible polygons, Util. Math. 58 (2000d), pp. 209-214.
%F For n >= 13, a(n) = A019434(n-7) (if it exists).
%t Select[Range[10^5],IntegerQ@Log2[EulerPhi@Binomial[#,2]]&] (* _Giorgos Kalogeropoulos_, Sep 08 2021 *)
%o (Magma) r:=7; IsInteger:=func<i | i eq Floor(i)>; lst:=[k: k in [2..6] | IsInteger(Log(2, EulerPhi(Binomial(k, 2))))]; t:=1; for x in [1..r] do m:=4^(2^x); if t eq 1 then Append(~lst, m); end if; if IsPrime(m+1) then Append(~lst, m+1); else t:=0; end if; end for; lst;
%Y Cf. A019434, A086700.
%K nonn,hard,more
%O 1,1
%A _Arkadiusz Wesolowski_, Aug 20 2021