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A371238
Euler totient function applied to the binary palindromes of even length.
1
2, 6, 8, 20, 24, 32, 36, 84, 96, 80, 108, 96, 144, 120, 128, 324, 320, 288, 420, 336, 360, 476, 384, 512, 432, 560, 540, 504, 632, 480, 600, 1364, 960, 1344, 1296, 1536, 1440, 1296, 1584, 1296, 1772, 1512, 1280, 1760, 1440, 1980, 1800, 1600, 1800, 2016, 1536, 1872
OFFSET
1,1
LINKS
William D. Banks and Igor E. Shparlinski, Average value of the Euler function on binary palindromes, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 54, No. 2 (2006), pp. 95-101; alternative link.
FORMULA
a(n) = A000010(A048701(n)).
(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).
MATHEMATICA
EulerPhi[Select[Range[5000], EvenQ[Length[(d = IntegerDigits[#, 2])]] && PalindromeQ[d] &]]
PROG
(PARI) is(n) = Vecrev(n = binary(n)) == n && !((#n)%2);
lista(kmax) = for(k = 1, kmax, if(is(k), print1(eulerphi(k), ", ")));
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Amiram Eldar, Mar 16 2024
STATUS
approved