%I #24 Jan 04 2022 09:22:49

%S 1,2,1,3,2,3,4,3,6,1,5,4,7,2,7,6,5,12,3,14,3,7,6,14,4,15,6,7,8,7,15,5,

%T 28,7,14,1,9,8,24,6,30,12,15,2,15,10,9,28,7,31,14,28,3,30,7,11,10,30,

%U 8,56,15,30,4,31,14,3,12,11,31,9,60,24,31,5,60,15,6,3,13,12,48,10,62,28,56,6,62,28,12,6,5,14,13,51,11,63,30,60,7,63,30,15,7,10,7

%N Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k), zero sequence (A000004) if no such solutions exist.

%C Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).

%C Square array is read by descending antidiagonals, as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

%C Rows at positions 2^k are 1, 2, 3, ..., (A000027). Row 2n is equal to row n.

%C Numbers on each row give a subset of positions of zeros at the corresponding row of A284270. - _Antti Karttunen_, May 08 2019

%H Antti Karttunen, <a href="/A115872/b115872.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array</a>

%H <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>

%e Fifteen initial terms of rows 1 - 19 are listed below:

%e 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

%e 2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

%e 3: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ...

%e 4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

%e 5: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...

%e 6: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ...

%e 7: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...

%e 8: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

%e 9: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...

%e 10: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...

%e 11: 3, 6, 12, 15, 24, 27, 30, 31, 48, 51, 54, 60, 62, 63, 96, ...

%e 12: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ...

%e 13: 5, 10, 15, 20, 21, 30, 31, 40, 42, 45, 47, 60, 61, 62, 63, ...

%e 14: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...

%e 15: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...

%e 16: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

%e 17: 31, 62, 63, 124, 126, 127, 248, 252, 254, 255, 496, 504, 508, 510, 511, ...

%e 18: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...

%e 19: 7, 14, 28, 31, 56, 62, 63, 112, 119, 124, 126, 127, 224, 238, 248, ...

%t X[a_, b_] := Module[{A, B, C, x},

%t A = Reverse@IntegerDigits[a, 2];

%t B = Reverse@IntegerDigits[b, 2];

%t C = Expand[

%t Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*

%t Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];

%t PolynomialMod[C, 2] /. x -> 2];

%t T[n_, k_] := Module[{x = BitXor[n-1, 2n-1], k0 = k},

%t For[i = 1, True, i++, If[n*i == X[x, i],

%t If[k0 == 1, Return[i], k0--]]]];

%t Table[T[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jan 04 2022 *)

%o (PARI)

%o up_to = 120;

%o A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);

%o A065621(n) = bitxor(n-1,n+n-1);

%o A115872sq(n, k) = { my(x = A065621(n)); for(i=1,oo,if((n*i)==A048720(x,i),if(1==k,return(i),k--))); };

%o A115872list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A115872sq(col,(a-(col-1))))); (v); };

%o v115872 = A115872list(up_to);

%o A115872(n) = v115872[n]; \\ (Slow) - _Antti Karttunen_, May 08 2019

%Y Transpose: A114388. First column: A115873.

%Y Cf. A048720, A065621, A115857, A115871, A325565, A325566, A325567, A325568, A325570, A325571.

%Y Cf. also arrays A277320, A277810, A277820, A284270.

%Y A few odd-positioned rows: row 1: A000027, Row 3: A048717, Row 5: A115770 (? Checked for all values less than 2^20), Row 7: A115770, Row 9: A115801, Row 11: A115803, Row 13: A115772, Row 15: A115801 (? Checked for all values less than 2^20), Row 17: A115809, Row 19: A115874, Row 49: A114384, Row 57: A114386.

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, Feb 07 2006

%E Example section added and the data section extended up to n=105 by _Antti Karttunen_, May 08 2019