A391726 Square array read by descending antidiagonals: find the k-th odd number i that satisfies i*n = A048720(i,m) for some m, then A(n, k) = m. Here A048720 is carryless base-2 multiplication.

1, 1, 2, 1, 2, 3, 1, 2, 7, 4, 1, 2, 3, 4, 5, 1, 2, 7, 4, 5, 6, 1, 2, 3, 4, 13, 14, 7, 1, 2, 7, 4, 5, 6, 11, 8, 1, 2, 3, 4, 13, 14, 7, 8, 9, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 1, 2, 7, 4, 5, 14, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 11, 8, 9, 26, 31, 12, 1, 2, 3, 4, 13, 6, 7, 8, 25, 10, 31, 28, 13

Offset

1,3

Comments

Array A391725 gives the corresponding k-th smallest odd i.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals

Index entries for sequences defined by congruent products between domains N and GF(2)[X].

Index entries for sequences related to binary expansion of n.

Index entries for sequences related to carryless arithmetic.

Formula

A(2*n, k) = 2 * A(n, k).

For all n, k: A048720(A(n,k), A391725(n,k)) = n * A391725(n,k).

For all n, k: A(n, k) >= n. [Implied by above, as x*y >= A048720(x,y)]

Example

The top left corner of the array:
n\k |  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16
----+----------------------------------------------------------------
1   |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
2   |  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,
3   |  3,  7,  3,  7,  3,  7,  3,  3,  7,  3,  3,  3,  7,  7,  3,  3,
4   |  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,
5   |  5,  5, 13,  5, 13,  5,  5,  5, 13,  5,  5,  5,  5, 13,  5,  5,
6   |  6, 14,  6, 14,  6, 14,  6,  6, 14,  6,  6,  6, 14, 14,  6,  6,
7   |  7, 11,  7, 11,  7, 11,  7, 11,  7,  7, 11,  7,  7,  7, 11,  7,
8   |  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,
9   |  9,  9,  9,  9, 25,  9,  9, 25,  9,  9,  9, 25,  9,  9,  9,  9,
10  | 10, 10, 26, 10, 26, 10, 10, 10, 26, 10, 10, 10, 10, 26, 10, 10,
11  | 11, 31, 31, 11, 31, 31, 11, 31, 31, 11, 31, 11, 31, 31, 31, 31,
12  | 12, 28, 12, 28, 12, 28, 12, 12, 28, 12, 12, 12, 28, 28, 12, 12,
13  | 13, 29, 21, 21, 13, 21, 21, 13, 21, 21, 21, 21, 13, 21, 21, 21,
14  | 14, 22, 14, 22, 14, 22, 14, 22, 14, 14, 22, 14, 14, 14, 22, 14,
15  | 15, 27, 23, 19, 15, 23, 19, 15, 27, 19, 15, 23, 19, 15, 27, 19,
16  | 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
17  | 17, 17, 17, 17, 17, 17, 17, 17, 49, 17, 17, 17, 17, 49, 17, 17,
18  | 18, 18, 18, 18, 50, 18, 18, 50, 18, 18, 18, 50, 18, 18, 18, 18,
19  | 19, 23, 19, 55, 55, 19, 55, 19, 55, 55, 19, 19, 19, 23, 55, 19,
20  | 20, 20, 52, 20, 52, 20, 20, 20, 52, 20, 20, 20, 20, 52, 20, 20,
21  | 21, 21, 29, 21, 29, 61, 21, 61, 21, 21, 21, 29, 21, 61, 21, 21,

Prog

(PARI)
up_to = 91;
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A391726_sq(n, k) = forstep(i=1, oo, 2, my(Pni=Pol(binary(n*i))*Mod(1, 2), P_i=Pol(binary(i))*Mod(1, 2)); if(0==lift(Pni % P_i), if(k>1, k--, return(fromdigits(Vec(lift(Pni / P_i)), 2)))));
A391726list(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] = A391726_sq(col, (a-(col-1))))); (v); };
v391726 = A391726list(up_to);
A391726(n) = v391726[n];

Crossrefs

Cf. A048720, A280500, A391725.

Columns 1-3: A005408, A391571, A391583.

Row 1: A000012.

Cf. also array A391736.

Sequence in context: A263714 A263703 A263752 * A101161 A245049 A214261

Adjacent sequences: A391723 A391724 A391725 * A391727 A391728 A391729

Keyword

nonn,base,tabl

Author

Antti Karttunen, Dec 18 2025

Status

approved