VOOZH about

URL: https://oeis.org/A391726

⇱ A391726 - OEIS

login
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.
7
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.
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
Columns 1-3: A005408, A391571, A391583.
Row 1: A000012.
Cf. also array A391736.
Sequence in context: A263714 A263703 A263752 * A101161 A245049 A214261
KEYWORD
nonn,base,tabl
AUTHOR
Antti Karttunen, Dec 18 2025
STATUS
approved