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A115872
Square array where row n gives all solutions k > 0 to the cross-domain congruence n*k = A048720(A065621(n),k).
30
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, 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, 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
OFFSET
1,2
COMMENTS
Here * stands for ordinary multiplication and X means carryless (GF(2)[X]) multiplication (A048720).
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.
Rows at positions 2^k are 1, 2, 3, ..., (A000027). Row 2n is equal to row n.
Numbers on each row give a subset of positions of zeros at the corresponding row of A284270. - Antti Karttunen, May 08 2019
Each row n is infinitely long, and contains as its initial term x = A115873(n) [which is well defined for all n, see comments there], and all its products of the form 2^e * x. It also contains y = A003817(n), at the index A391729(n), and all its products of the form 2^e * y. If A391729(n) = 1, then x and y are equal. - Antti Karttunen, Dec 20 2025
This is a subarray of more extensive A391925, where each row n gives all solutions i to i*n = A048720(i,m) for some m. It is guaranteed that A065621(n) is always one of these m. - Antti Karttunen, Dec 23 2025
FORMULA
A(2*n, k) = A(n, k).
For all n >= 1, A(n, A391729(n)) = A003817(n).
EXAMPLE
Fifteen initial terms of rows 1 - 19 are listed below:
1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
3: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ...
4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
5: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...
6: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ...
7: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...
8: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
9: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
10: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...
11: 3, 6, 12, 15, 24, 27, 30, 31, 48, 51, 54, 60, 62, 63, 96, ...
12: 3, 6, 7, 12, 14, 15, 24, 28, 30, 31, 48, 51, 56, 60, 62, ...
13: 5, 10, 15, 20, 21, 30, 31, 40, 42, 45, 47, 60, 61, 62, 63, ...
14: 7, 14, 15, 28, 30, 31, 56, 60, 62, 63, 112, 120, 124, 126, 127, ...
15: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
16: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
17: 31, 62, 63, 124, 126, 127, 248, 252, 254, 255, 496, 504, 508, 510, 511, ...
18: 15, 30, 31, 60, 62, 63, 120, 124, 126, 127, 240, 248, 252, 254, 255, ...
19: 7, 14, 28, 31, 56, 62, 63, 112, 119, 124, 126, 127, 224, 238, 248, ...
MATHEMATICA
X[a_, b_] := Module[{A, B, C, x},
A = Reverse@IntegerDigits[a, 2];
B = Reverse@IntegerDigits[b, 2];
C = Expand[
Sum[A[[i]]*x^(i-1), {i, 1, Length[A]}]*
Sum[B[[i]]*x^(i-1), {i, 1, Length[B]}]];
PolynomialMod[C, 2] /. x -> 2];
T[n_, k_] := Module[{x = BitXor[n-1, 2n-1], k0 = k},
For[i = 1, True, i++, If[n*i == X[x, i],
If[k0 == 1, Return[i], k0--]]]];
Table[T[n-k+1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2022 *)
PROG
(PARI)
up_to = 120;
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1, n+n-1);
A115872sq(n, k) = { my(x = A065621(n)); for(i=1, oo, if((n*i)==A048720(x, i), if(1==k, return(i), k--))); };
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); };
v115872 = A115872list(up_to);
A115872(n) = v115872[n]; \\ (Slow) - Antti Karttunen, May 08 2019
CROSSREFS
Transpose: A114388. First column: A115873.
Cf. also arrays A277320, A277810, A277820, A284270, A391925.
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.
Sequence in context: A240450 A352129 A340351 * A133926 A144337 A143929
KEYWORD
nonn,base,tabl
AUTHOR
Antti Karttunen, Feb 07 2006
EXTENSIONS
Example section added and the data section extended up to n=105 by Antti Karttunen, May 08 2019
Unnecessary escape-clause removed from the definition by Antti Karttunen, Dec 20 2025
STATUS
approved