A354059 - OEIS

%I #16 Jul 06 2024 19:45:24

%S 1,1,0,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,0,1,1,0,2,0,0,

%T 0,0,1,3,2,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,1,2,0,0,0,0,0,0,0,0,1,1,

%U 0,0,0,2,0,0,0,0,0,0,1,3,0,2,0,0,0,0,0,0,0,0,0

%N Square array read by ascending antidiagonals: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k.

%C Row n and Row n' are the same if and only if (Z/nZ)* = (Z/n'Z)*, where (Z/nZ)* is the multiplicative group of integers modulo n.

%C For the truncated version see A252911.

%H Jianing Song, <a href="/A354059/b354059.txt">Table of n, a(n) for n = 1..5050</a>

%F A327924(n,k) = Sum_{d|k} T(n,k)/phi(d).

%e The 7th, 9th, 14th and 18th rows of A354047 are {1,2,3,2,1,6,1,2,3,2,1,6,...}, so applying the Moebius transform gives {1,1,2,0,0,2,0,0,0,0,0,0,...}.

%o (PARI) b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));

%o T(n,k) = sumdiv(k, d, moebius(k/d)*b(n,d))

%Y Moebius transform of A354057 applied to each row.

%Y Cf. A327924.

%K nonn,tabl

%O 1,32

%A _Jianing Song_, May 16 2022