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 A370050 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. 10
 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, 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, 1, 2, 1, 2, 1, 2, 3, 10, 1, 8, 1, 2, 3, 4, 1, 6, 1, 4, 1, 2, 1, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. Each row is multiplicative. LINKS Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals) FORMULA 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). EXAMPLE Table reads 1, 4, 1, 8, 1, 4, 1, 16, 1, 4 1, 2, 3, 2, 1, 6, 1, 2, 9, 2 1, 2, 1, 4, 5, 2, 1, 4, 1, 10 1, 2, 3, 2, 1, 6, 7, 2, 3, 2 1, 2, 1, 2, 5, 2, 1, 2, 1, 10 1, 2, 3, 4, 1, 6, 1, 4, 3, 2 1, 2, 1, 4, 1, 2, 1, 8, 1, 2 1, 2, 3, 2, 1, 6, 1, 2, 9, 2 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 1, 2, 1, 4, 1, 2, 7, 4, 1, 2 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. 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. PROG (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) CROSSREFS Cf. A370067. Row 1-4: A297402, A370180, A370181, A370182. Sequence in context: A108536 A232631 A153094 * A144870 A370211 A256252 Adjacent sequences: A370047 A370048 A370049 * A370051 A370052 A370053 KEYWORD nonn,tabl,easy AUTHOR Jianing Song, Apr 30 2024 STATUS approved

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Last modified August 11 11:30 EDT 2024. Contains 375068 sequences. (Running on oeis4.)