A371238 Euler totient function applied to the binary palindromes of even length.

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

Author

Amiram Eldar, Mar 16 2024

Keywords

• nonn
• base
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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.

Crossrefs

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

Programs

Mathematica

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

PARI

is(n) = Vecrev(n = binary(n)) == n && !((#n)%2);
lista(kmax) = for(k = 1, kmax, if(is(k), print1(eulerphi(k), ", ")));

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).