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 A354057 #25 Jul 07 2024 01:46:07
%S A354057 1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,1,2,1,4,1,2,
%T A354057 1,1,1,4,3,2,1,2,1,1,1,1,2,1,2,1,2,1,2,1,1,1,2,3,4,1,2,1,2,1,1,1,1,2,
%U A354057 1,2,1,6,1,4,1,2,1,1,1,4,1,4,1,4,1,2,1,2,1,1,1
%N A354057 Square array read by ascending antidiagonals: T(n,k) is the number of solutions to x^k == 1 (mod n).
%C A354057 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 A354057 Given n, T(n,k) only depends on gcd(k,psi(n)). For the truncated version see A354060.
%C A354057 Each column is multiplicative.
%H A354057 Jianing Song, <a href="/A354057/b354057.txt">Table of n, a(n) for n = 1..5050</a> (the first 100 ascending diagonals)
%F A354057 If (Z/nZ)* = C_{k_1} X C_{k_2} X ... X C_{k_r}, then T(n,k) = Product_{i=1..r} gcd(k,k_r).
%F A354057 T(p^e,k) = gcd((p-1)*p^(e-1),k) for odd primes p. T(2,k) = 1, T(2^e,k) = 2*gcd(2^(e-2),k) if k is even and 1 if k is odd.
%F A354057 A327924(n,k) = Sum_{q|n} T(n,k) * (Sum_{s|n/q} mu(s)/phi(s*q)).
%e A354057 n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
%e A354057 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A354057 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A354057 3 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
%e A354057 4 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
%e A354057 5 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4
%e A354057 6 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
%e A354057 7 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
%e A354057 8 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4
%e A354057 9 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
%e A354057 10 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4
%e A354057 11 1 2 1 2 5 2 1 2 1 10 1 2 1 2 5 2 1 2 1 10
%e A354057 12 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4
%e A354057 13 1 2 3 4 1 6 1 4 3 2 1 12 1 2 3 4 1 6 1 4
%e A354057 14 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
%e A354057 15 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8
%e A354057 16 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8
%e A354057 17 1 2 1 4 1 2 1 8 1 2 1 4 1 2 1 16 1 2 1 4
%e A354057 18 1 2 3 2 1 6 1 2 3 2 1 6 1 2 3 2 1 6 1 2
%e A354057 19 1 2 3 2 1 6 1 2 9 2 1 6 1 2 3 2 1 18 1 2
%e A354057 20 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8
%o A354057 (PARI) T(n,k)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(k, Z[i]))
%Y A354057 k-th column: A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), A319101 (k=7), A247257 (k=8).
%Y A354057 Applying Moebius transform to the rows gives A354059.
%Y A354057 Applying Moebius transform to the columns gives A354058.
%Y A354057 Cf. A327924.
%K A354057 nonn,tabl
%O A354057 1,8
%A A354057 _Jianing Song_, May 16 2022