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A370067 Square array read by ascending antidiagonals: T(n,k) is the size of the group Q_p*/(Q_p*)^k, where p = prime(n), and Q_p is the field of p-adic numbers.

%I A370067 #13 May 12 2024 17:58:21
%S A370067 1,1,8,1,4,3,1,4,9,32,1,4,3,8,5,1,4,9,16,5,24,1,4,3,8,25,36,7,1,4,9,8,
%T A370067 5,12,7,128,1,4,3,16,25,36,7,16,9,1,4,9,16,5,12,49,32,81,40,1,4,3,8,5,
%U A370067 36,7,16,9,20,11,1,4,3,8,5,12,7,16,27,100,11,96,1,4,9,16,5,36,7,32,9,20,11,72,13
%N A370067 Square array read by ascending antidiagonals: T(n,k) is the size of the group Q_p*/(Q_p*)^k, where p = prime(n), and Q_p is the field of p-adic numbers.
%C A370067 We have Q_p* = p^Z X Z_p*, so Q_p*/(Q_p*)^k = (p^Z/p^(kZ)) X (Z_p*/(Z_p*)^k). Note that p^Z/p^(kZ) is a cyclic group of order k. For the group structure of (Z_p*/(Z_p*)^k), see A370050.
%C A370067 Each row is multiplicative.
%H A370067 Jianing Song, <a href="/A370067/b370067.txt">Table of n, a(n) for n = 1..5050</a> (first 100 antidiagonals)
%F A370067 T(n,k) = k * A370050(n,k).
%F A370067 Write k = p^e * k' with k' not being divisible by p, and p = prime(n). If p is odd, then T(n,k) = k * p^e * gcd(p-1,k'). If p = 2 and k is odd, then T(n,k) = k. If p = 2 and k is even, then T(n,k) = k * 2^(e+1).
%e A370067 Table reads
%e A370067   1, 8, 3, 32, 5, 24, 7, 128, 9, 40
%e A370067   1, 4, 9, 8, 5, 36, 7, 16, 81, 20
%e A370067   1, 4, 3, 16, 25, 12, 7, 32, 9, 100
%e A370067   1, 4, 9, 8, 5, 36, 49, 16, 27, 20
%e A370067   1, 4, 3, 8, 25, 12, 7, 16, 9, 100
%e A370067   1, 4, 9, 16, 5, 36, 7, 32, 27, 20
%e A370067   1, 4, 3, 16, 5, 12, 7, 64, 9, 20
%e A370067   1, 4, 9, 8, 5, 36, 7, 16, 81, 20
%e A370067   1, 4, 3, 8, 5, 12, 7, 16, 9, 20
%e A370067   1, 4, 3, 16, 5, 12, 49, 32, 9, 20
%o A370067 (PARI) T(n, k) = my(p = prime(n), e = valuation(k, p)); k * p^e*gcd(p-1, k/p^e) * if(p==2 && e>=1, 2, 1)
%Y A370067 Cf. A370050. Row 1-4: A370564, A370565, A370566, A370567.
%K A370067 nonn,tabl,easy
%O A370067 1,3
%A A370067 _Jianing Song_, Apr 30 2024