This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060367 #29 Jul 25 2025 06:46:51
%S A060367 1,1,2,2,4,3,6,5,6,6,10,6,12,9,9,10,16,10,18,11,14,15,22,12,20,18,20,
%T A060367 16,28,14,30,21,23,24,25,18,36,27,28,22,40,21,42,27,28,33,46,24,42,31,
%U A060367 37,33,52,30,42,33,42,42,58,26,60,45,41,42,50,35,66,44,51
%N A060367 Average order of an element in a cyclic group of order n rounded down.
%H A060367 Danny Rorabaugh, <a href="/A060367/b060367.txt">Table of n, a(n) for n = 1..10000</a>
%F A060367 Sequence A057660 gives the sum of the orders of the elements in a cyclic group with n elements so a(n) = floor(A057660(n) / n) = floor(Sum_{k=1..n} 1/GCD(n, k)) = floor(Sum of 1/d times phi(n/d)) for all divisors d of n, where phi is Euler's phi function. This sum may also be expressed as the product of (p^(2*e(p)+1)+1)/((p+1)*p^e(p)) over all prime divisors p of n where the canonical factorization of n is the product of p^e(p), the e(p) being the exponents of the power of p in the factorization.
%F A060367 From _Amiram Eldar_, Jul 25 2025: (Start)
%F A060367 a(n) = floor(sigma_2(n^2)/(n*sigma(n))) = floor(A001157(n^2)/(n*A000203(n^2))) = floor(A065827(n)/(n*A065764(n))).
%F A060367 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(3)/zeta(2) (A253905). (End)
%p A060367 seq(floor(numtheory:-sigma[2](n^2)/numtheory:-sigma(n^2)/n), n=1..1000); # _Robert Israel_, Mar 24 2015
%t A060367 f[n_] := Block[{i, j, k}, Reap@ For[j = 1, j <= n, j++, Sow[Floor[Sum[1/GCD[j, k], {k, 1, j}]]]]] // Flatten // Rest; f@ 49 (* _Michael De Vlieger_, Mar 24 2015 *)
%t A060367 a[n_] := Floor[DivisorSigma[2, n^2]/DivisorSigma[1, n^2]/n]; Array[a, 100] (* _Amiram Eldar_, Jul 25 2025 *)
%o A060367 (Sage) [floor(sum([1/gcd(n,k) for k in range(1,n+1)])) for n in range(1,50)] # _Danny Rorabaugh_, Mar 24 2015
%o A060367 (PARI) a(n) = {my(f = factor(n^2)); floor(sigma(f, 2)/(n * sigma(f)));} \\ _Amiram Eldar_, Jul 25 2025
%Y A060367 Cf. A000010, A000203, A001157, A018804, A057660, A065764, A065827, A253905.
%K A060367 nonn,easy
%O A060367 1,3
%A A060367 Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
%E A060367 Offset corrected and terms a(18)-a(50) added by _Danny Rorabaugh_, Mar 24 2015