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A354061 Irregular table read by rows: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n, 1 <= k <= psi(n), psi = A002322.

%I A354061 #23 May 25 2022 03:35:42
%S A354061 1,0,0,1,0,1,0,1,0,3,0,0,0,1,2,1,0,5,0,2,0,0,2,0,0,4,0,0,0,0,0,1,0,1,
%T A354061 4,1,0,1,0,9,0,1,0,1,2,3,0,5,0,3,2,1,0,11,0,0,0,0,0,0,0,1,0,3,0,0,0,4,
%U A354061 0,1,0,3,0,1,0,7,0,1,0,3,0,1,0,15
%N A354061 Irregular table read by rows: T(n,k) is the number of degree-k primitive Dirichlet characters modulo n, 1 <= k <= psi(n), psi = A002322.
%C A354061 Given n, T(n,k) only depends on gcd(k,psi(n)).
%C A354061 The n-th row contains entirely 0's if and only if n == 2 (mod 4).
%C A354061 If n !== 2 (mod 4), T(n,psi(n)) > T(n,k) for 1 <= k < psi(n).
%H A354061 Jianing Song, <a href="/A354061/b354061.txt">Table of n, a(n) for n = 1..8346</a> (the first 200 rows)
%F A354061 For odd primes p: T(p,k) = gcd(p-1,k)-1, T(p^e,k*p^(e-1)) = p^(e-2)*(p-1)*gcd(k,p-1), T(p^e,k) = 0 if k is not divisible by p^(e-1). T(2,k) = 0, T(4,k) = 1 for even k and 0 for odd k, T(2^e,k) = 2^(e-2) if k is divisible by 2^(e-2) and 0 otherwise.
%F A354061 T(n,psi(n)) = A007431(n). - _Jianing Song_, May 24 2022
%e A354061 Table starts
%e A354061 n = 1: 1;
%e A354061 n = 2: 0;
%e A354061 n = 3: 0, 1;
%e A354061 n = 4: 0, 1;
%e A354061 n = 5: 0, 1, 0, 3;
%e A354061 n = 6: 0, 0;
%e A354061 n = 7: 0, 1, 2, 1, 0, 5;
%e A354061 n = 8: 0, 2;
%e A354061 n = 9: 0, 0, 2, 0, 0, 4;
%e A354061 n = 10: 0, 0, 0, 0;
%e A354061 n = 11: 0, 1, 0, 1, 4, 1, 0, 1, 0, 9;
%e A354061 n = 12: 0, 1;
%e A354061 n = 13: 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11;
%e A354061 n = 14: 0, 0, 0, 0, 0, 0;
%e A354061 n = 15: 0, 1, 0, 3;
%e A354061 n = 16: 0, 0, 0, 4;
%e A354061 n = 17: 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 15;
%e A354061 n = 18: 0, 0, 0, 0, 0, 0;
%e A354061 n = 19: 0, 1, 2, 1, 0, 5, 0, 1, 8, 1, 0, 5, 0, 1, 2, 1, 0, 17;
%e A354061 n = 20: 0, 1, 0, 3;
%e A354061 ...
%o A354061 (PARI) b(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]));
%o A354061 T(n,k) = sumdiv(n, d, moebius(n/d)*b(d,k))
%Y A354061 Cf. A354058, A002322, A007431, A354060.
%Y A354061 A354257 gives the smallest index for the nonzero terms in each row.
%K A354061 nonn,tabf
%O A354061 1,10
%A A354061 _Jianing Song_, May 16 2022