A391735 Square array read by descending antidiagonals: A(n, k) is the k-th odd number i such that i divides A048720(n,i), where A048720 is carryless base-2 multiplication.

1, 3, 1, 5, 3, 1, 7, 5, 5, 1, 9, 7, 9, 3, 1, 11, 9, 17, 5, 3, 1, 13, 11, 21, 7, 9, 5, 1, 15, 13, 33, 9, 17, 9, 3, 1, 17, 15, 37, 11, 19, 17, 7, 3, 1, 19, 17, 41, 13, 25, 21, 9, 5, 3, 1, 21, 19, 65, 15, 33, 33, 15, 7, 5, 3, 1, 23, 21, 69, 17, 35, 37, 17, 9, 7, 9, 7, 1, 25, 23, 73, 19, 49, 41, 31, 11, 17, 17, 15, 5, 1

Offset

1,2

Comments

Array A391736 gives the corresponding m.

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) = A(n, k).

A(n,k) * A391736(n,k) = A048720(A(n,k), n).

Example

The top left corner of the array:
n\k | 1   2   3   4    5    6    7    8    9   10    11    12    13    14
----+---------------------------------------------------------------------
1   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,
2   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,
3   | 1,  5,  9, 17,  21,  33,  37,  41,  65,  69,   73,   81,   85,  129,
4   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,
5   | 1,  3,  9, 17,  19,  25,  33,  35,  49,  51,   65,   67,   73,   97,
6   | 1,  5,  9, 17,  21,  33,  37,  41,  65,  69,   73,   81,   85,  129,
7   | 1,  3,  7,  9,  15,  17,  31,  33,  51,  63,   65,   73,   99,  103,
8   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,
9   | 1,  3,  5,  7,  17,  21,  33,  35,  49,  65,   67,   69,   71,   81,
10  | 1,  3,  9, 17,  19,  25,  33,  35,  49,  51,   65,   67,   73,   97,
11  | 1,  7, 15, 17,  31,  33,  63,  65, 127, 129,  255,  257,  273,  455,
12  | 1,  5,  9, 17,  21,  33,  37,  41,  65,  69,   73,   81,   85,  129,
13  | 1,  7, 15, 17,  31,  33,  63,  65, 127, 129,  255,  257,  273,  455,
14  | 1,  3,  7,  9,  15,  17,  31,  33,  51,  63,   65,   73,   99,  103,
15  | 1, 17, 33, 65, 129, 257, 273, 513, 529, 545, 1025, 1041, 1057, 1089,
16  | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,
17  | 1,  3,  5,  7,   9,  11,  13,  15,  33,  37,   41,   45,   65,   67,
18  | 1,  3,  5,  7,  17,  21,  33,  35,  49,  65,   67,   69,   71,   81,
19  | 1,  5, 15, 31,  33,  63,  65, 127, 129, 133,  161,  255,  257,  261,
20  | 1,  3,  9, 17,  19,  25,  33,  35,  49,  51,   65,   67,   73,   97,
21  | 1,  3,  5,  9,  15,  21,  31,  33,  45,  47,   61,   63,   65,   67,

Prog

(PARI)
up_to = 105;
A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A391735_sq(n, k) = forstep(i=1, oo, 2, if(!(A048720(n, i)%i), if(k>1, k--, return(i))));
A391735list(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] = A391735_sq(col, (a-(col-1))))); (v); };
v391735 = A391735list(up_to);
A391735(n) = v391735[n];

Crossrefs

Cf. A048720, A391736.

Row 1: A005408.

Column 2: A391572.

Cf. also A391725.

Sequence in context: A099375 A391725 A348835 * A130301 A133601 A258207

Adjacent sequences: A391732 A391733 A391734 * A391736 A391737 A391738

Keyword

nonn,base,tabl

Author

Antti Karttunen, Dec 18 2025

Status

approved