A371238 - OEIS

%I #8 Mar 16 2024 04:19:43

%S 2,6,8,20,24,32,36,84,96,80,108,96,144,120,128,324,320,288,420,336,

%T 360,476,384,512,432,560,540,504,632,480,600,1364,960,1344,1296,1536,

%U 1440,1296,1584,1296,1772,1512,1280,1760,1440,1980,1800,1600,1800,2016,1536,1872

%N Euler totient function applied to the binary palindromes of even length.

%H Amiram Eldar, <a href="/A371238/b371238.txt">Table of n, a(n) for n = 1..10000</a>

%H William D. Banks and Igor E. Shparlinski, <a href="http://dx.doi.org/10.4064/ba54-2-1">Average value of the Euler function on binary palindromes</a>, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 54, No. 2 (2006), pp. 95-101; <a href="https://eudml.org/doc/286495">alternative link</a>.

%F a(n) = A000010(A048701(n)).

%F (1/N(k)) * Sum_{j, A070939(A048701(j)) = 2*k} a(j) = 3 * 2^(2*k-2) * (6/Pi^2 + O((k/log(k))^(-1/4))), where N(k) = Sum_{j, A070939(A048701(j)) = 2*k} 1 (Banks and Shparlinski, 2006).

%t EulerPhi[Select[Range[5000], EvenQ[Length[(d = IntegerDigits[#, 2])]] && PalindromeQ[d] &]]

%o (PARI) is(n) = Vecrev(n = binary(n)) == n && !((#n)%2);

%o lista(kmax) = for(k = 1, kmax, if(is(k), print1(eulerphi(k), ", ")));

%Y Cf. A000010, A006995, A048701, A059956, A070939.

%K nonn,base,easy

%O 1,1

%A _Amiram Eldar_, Mar 16 2024