A391736 Square array read by descending antidiagonals: A(n, k) is the value of quotient A048720(n,i)/i for the k-th odd number i which divides A048720(n,i), where A048720 is carryless base-2 multiplication.

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 3, 8, 1, 2, 3, 4, 5, 6, 3, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 3, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 12, 1, 2, 3, 4, 5, 6, 3, 8, 9, 10, 7, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 14

Offset

1,3

Comments

Array A391735 gives the corresponding odd numbers i that satisfy the requirement.

The number of distinct values that may occur on row n of this array is at least 1+A115861(n), and at most n, because A(n, k) <= n. We conjecture that 1+A115861(n) gives the exact value.

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: A391736(n,k) * A391735(n,k) = A048720(n, A391735(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  17
----+--------------------------------------------------------------------
1   |  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,  2,
3   |  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,
4   |  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,
5   |  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,  5,
6   |  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,
7   |  7,  3,  3,  7,  3,  7,  3,  7,  3,  3,  7,  7,  3,  3,  3,  3,  7,
8   |  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,
9   |  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,  9,
10  | 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
11  | 11,  7,  7, 11,  7, 11,  7, 11,  7, 11,  7, 11, 11,  7,  7, 11, 11,
12  | 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12,
13  | 13,  5,  5, 13,  5, 13,  5, 13,  5, 13,  5, 13, 13,  5,  5, 13, 13,
14  | 14,  6,  6, 14,  6, 14,  6, 14,  6,  6, 14, 14,  6,  6,  6,  6, 14,
15  | 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,
16  | 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, 17, 17, 17, 17, 17, 17, 17, 17, 17,
18  | 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18,
19  | 19, 19, 15, 15, 19, 15, 19, 15, 19, 19, 19, 15, 19, 19, 19, 15, 19,
20  | 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20,
21  | 21, 21, 13, 21, 13, 13, 13, 21, 13, 13, 13, 13, 21, 21, 21, 13, 13,

Prog

(PARI)
up_to = 105;
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A391736_sq(n, k) = forstep(i=1, oo, 2, if(!(A048720(n, i)%i), if(k>1, k--, return(A048720(n, i)/i))));
A391736list(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] = A391736_sq(col, (a-(col-1))))); (v); };
v391736 = A391736list(up_to);
A391736(n) = v391736[n];

Crossrefs

Cf. A115861, A391735.

Row 1: A000012.

Column 2: A391573.

Cf. also A391726.

Sequence in context: A348043 A138060 A023121 * A136261 A140756 A002260

Adjacent sequences: A391733 A391734 A391735 * A391737 A391738 A391739

Keyword

nonn,base,tabl

Author

Antti Karttunen, Dec 18 2025

Status

approved