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A354059 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.

%I A354059 #16 Jul 06 2024 19:45:24
%S A354059 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 A354059 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 A354059 0,0,0,2,0,0,0,0,0,0,1,3,0,2,0,0,0,0,0,0,0,0,0
%N A354059 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 A354059 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 A354059 For the truncated version see A252911.
%H A354059 Jianing Song, <a href="/A354059/b354059.txt">Table of n, a(n) for n = 1..5050</a>
%F A354059 A327924(n,k) = Sum_{d|k} T(n,k)/phi(d).
%e A354059 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 A354059 (PARI) b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
%o A354059 T(n,k) = sumdiv(k, d, moebius(k/d)*b(n,d))
%Y A354059 Moebius transform of A354057 applied to each row.
%Y A354059 Cf. A327924.
%K A354059 nonn,tabl
%O A354059 1,32
%A A354059 _Jianing Song_, May 16 2022