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Square array read by ascending antidiagonals: T(n,k) is the size of the group Z_p*/(Z_p*)^k, where p = prime(n), and Z_p is the ring of p-adic integers.
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%I #15 May 12 2024 17:58:41

%S 1,1,4,1,2,1,1,2,3,8,1,2,1,2,1,1,2,3,4,1,4,1,2,1,2,5,6,1,1,2,3,2,1,2,

%T 1,16,1,2,1,4,5,6,1,2,1,1,2,3,4,1,2,7,4,9,4,1,2,1,2,1,6,1,2,1,2,1,1,2,

%U 1,2,1,2,1,2,3,10,1,8,1,2,3,4,1,6,1,4,1,2,1,6,1

%N Square array read by ascending antidiagonals: T(n,k) is the size of the group Z_p*/(Z_p*)^k, where p = prime(n), and Z_p is the ring of p-adic integers.

%C We have that Z_p*/(Z_p*)^k is the inverse limit of (Z/p^iZ)*/((Z/p^iZ)*)^k as i tends to infinity. Write k = p^e * k' with k' not being divisible by p. If p is odd, then the group is cyclic of order p^e * gcd(p-1,k'). If p = 2 and k is odd, then the group is trivial. If p = 2 and k is even, then the group is the product of a cyclic group of order 2^e and a cyclic group of order 2.

%C Each row is multiplicative.

%H Jianing Song, <a href="/A370050/b370050.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals)

%F Write k = p^e * k' with k' not being divisible by p, and p = prime(n). If p is odd, then T(n,k) = p^e * gcd(p-1,k'). If p = 2 and k is odd, then T(n,k) = 1. If p = 2 and k is even, then T(n,k) = 2^(e+1).

%e Table reads

%e 1, 4, 1, 8, 1, 4, 1, 16, 1, 4

%e 1, 2, 3, 2, 1, 6, 1, 2, 9, 2

%e 1, 2, 1, 4, 5, 2, 1, 4, 1, 10

%e 1, 2, 3, 2, 1, 6, 7, 2, 3, 2

%e 1, 2, 1, 2, 5, 2, 1, 2, 1, 10

%e 1, 2, 3, 4, 1, 6, 1, 4, 3, 2

%e 1, 2, 1, 4, 1, 2, 1, 8, 1, 2

%e 1, 2, 3, 2, 1, 6, 1, 2, 9, 2

%e 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

%e 1, 2, 1, 4, 1, 2, 7, 4, 1, 2

%e For p = prime(1) = 2 and k = 2, we have Z_p*/(Z_p*)^k = Z_2*/(1+8Z_2) = (Z/8Z)*/(1+8Z) = C_2 X C_2, so T(1,2) = 4.

%e For p = prime(2) = 3 and k = 3, we have Z_p*/(Z_p*)^k = Z_3*/((1+9Z_3) U (8+9Z_3)) = (Z/9Z)*/((1+9Z) U (8+9Z)) = C_3, so T(2,3) = 3.

%o (PARI) T(n,k) = my(p = prime(n), e = valuation(k,p)); p^e*gcd(p-1,k/p^e) * if(p==2 && e>=1, 2, 1)

%Y Cf. A370067. Row 1-4: A297402, A370180, A370181, A370182.

%K nonn,tabl,easy

%O 1,3

%A _Jianing Song_, Apr 30 2024