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A321906
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Irregular table read by rows: T(n,k) is the smallest m such that m^(-1/m) == 2*k + 1 (mod 2^n), 0 <= k <= 2^(n-1) - 1.
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7
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1, 1, 3, 1, 3, 5, 7, 1, 3, 13, 7, 9, 11, 5, 15, 1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31, 1, 19, 61, 7, 57, 27, 53, 15, 49, 35, 45, 23, 41, 43, 37, 31, 33, 51, 29, 39, 25, 59, 21, 47, 17, 3, 13, 55, 9, 11, 5, 63
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OFFSET
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1,3
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COMMENTS
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T(n,k) is the unique x in {1, 3, 5, ..., 2^n - 1} such that (2*k+1)^(-m) == m (mod 2^n).
The n-th row contains 2^(n-1) numbers, and is a permutation of the odd numbers below 2^n.
For all n, k we have v(T(n,k)-1, 2) = v(k, 2) + 1 and v(T(n,k)+1, 2) = v(k+1, 2) + 1, where v(k, 2) = A007814(k) is the 2-adic valuation of k.
For n >= 3, T(n,k) = 2*k + 1 iff k == -1 (mod 2^floor((n-1)/2)) or k = 0 or k = 2^(n-2).
T(n,k) is the multiplicative inverse of A320562(n,k) modulo 2^n.
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LINKS
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FORMULA
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T(n,k) = 2^n - A321905(n,2^(n-1)-1-k).
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EXAMPLE
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Table starts
1,
1, 3,
1, 3, 5, 7,
1, 3, 13, 7, 9, 11, 5, 15,
1, 19, 29, 7, 25, 27, 21, 15, 17, 3, 13, 23, 9, 11, 5, 31,
1, 19, 61, 7, 57, 27, 53, 15, 49, 35, 45, 23, 41, 43, 37, 31, 33, 51, 29, 39, 25, 59, 21, 47, 17, 3, 13, 55, 9, 11, 5, 63,
...
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PROG
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(PARI) T(n, k) = my(m=1); while(Mod(2*k+1, 2^n)^(-m)!=m, m+=2); m
tabf(nn) = for(n=1, nn, for(k=0, 2^(n-1)-1, print1(T(n, k), ", ")); print)
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CROSSREFS
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{x^(-1/x)} and its inverse: A321903 & this sequence.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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