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Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation appears as a substring in the binary representation of 1/n (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).
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%I #7 Mar 11 2018 17:17:45

%S 1,1,2,1,2,4,1,2,4,8,1,2,5,8,16,1,2,4,10,16,32,1,2,3,8,21,32,64,1,2,5,

%T 4,16,42,64,128,1,2,4,10,6,32,85,128,256,1,2,4,9,21,9,64,170,256,512,

%U 1,2,3,8,18,42,12,128,341,512,1024,1,2,3,4,16,36,85

%N Square array T(n, k) (n >= 1, k >= 1) read by antidiagonals upwards: T(n, k) is the k-th positive number whose binary representation appears as a substring in the binary representation of 1/n (ignoring the radix point and adding trailing zeros if necessary in case of a terminating expansion).

%H Rémy Sigrist, <a href="/A300653/a300653.gp.txt">PARI program for A300653</a>

%F T(n, 1) = 1.

%F T(n, 2) = 2.

%F T(n, 3) > 3 iff n belongs to A300630.

%F T(2*n, k) = T(n, k).

%F T(1, k) = A000079(k-1).

%F T(3, k) = A000975(k).

%F T(7, k) = A033138(k).

%F T(n, k) = k iff 1 <= k <= A300654(n).

%F T(n, k) = n for some k iff n belongs to A000079 or to A153055.

%e Square array begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10

%e ---+--------------------------------------------------

%e 1| 1 2 4 8 16 32 64 128 256 512 --> A000079

%e 2| 1 2 4 8 16 32 64 128 256 512

%e 3| 1 2 5 10 21 42 85 170 341 682 --> A000975

%e 4| 1 2 4 8 16 32 64 128 256 512

%e 5| 1 2 3 4 6 9 12 19 25 38

%e 6| 1 2 5 10 21 42 85 170 341 682

%e 7| 1 2 4 9 18 36 73 146 292 585 --> A033138

%e 8| 1 2 4 8 16 32 64 128 256 512

%e 9| 1 2 3 4 6 7 8 12 14 17

%e 10| 1 2 3 4 6 9 12 19 25 38

%o (PARI) See Links section.

%Y Cf. A000079, A000975, A033138, A153055, A300630, A300654.

%K nonn,base,tabl

%O 1,3

%A _Rémy Sigrist_, Mar 10 2018