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 A082953 #33 Jun 04 2025 11:30:47
%S A082953 1,2,4,8,16,8,36,32,36,32,100,32,144,72,64,128,256,72,324,128,144,200,
%T A082953 484,128,400,288,324,288,784,128,900,512,400,512,576,288,1296,648,576,
%U A082953 512,1600,288,1764,800,576,968,2116
%N A082953 a(n) = A000252(n) / A070732(n).
%C A082953 From _Jianing Song_, Apr 20 2019: (Start)
%C A082953 a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
%C A082953 a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)
%H A082953 Michael De Vlieger, <a href="/A082953/b082953.txt">Table of n, a(n) for n = 1..10000</a>
%F A082953 a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - _Vladeta Jovovic_, May 02 2005
%F A082953 Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - _R. J. Mathar_, Apr 14 2011
%F A082953 a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - _Jianing Song_, Apr 20 2019
%F A082953 Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 = 0.171299... . - _Amiram Eldar_, Oct 30 2022
%F A082953 a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - _Ridouane Oudra_, Jun 04 2025
%p A082953 A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
%p A082953 seq(A082953(n),n=1..100) ; # _R. J. Mathar_, Jan 07 2011
%t A082953 Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* _Michael De Vlieger_, Apr 21 2019 *)
%o A082953 (PARI) a(n) = eulerphi(n)*eulerphi(2*n); \\ _Michel Marcus_, Jun 04 2025
%Y A082953 Cf. A000252, A065464, A070732.
%Y A082953 Cf. A000010, A062570, A040001.
%Y A082953 Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).
%K A082953 mult,nonn
%O A082953 1,2
%A A082953 Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003