A391735 - OEIS

%I #7 Dec 18 2025 22:24:24

%S 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,

%T 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,

%U 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

%N 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.

%C Array A391736 gives the corresponding m.

%H Antti Karttunen, <a href="/A391735/b391735.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals</a>

%H <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>.

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

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

%e The top left corner of the array:

%e n\k | 1   2   3   4    5    6    7    8    9   10    11    12    13    14

%e ----+---------------------------------------------------------------------

%e 1   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,

%e 2   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,

%e 3   | 1,  5,  9, 17,  21,  33,  37,  41,  65,  69,   73,   81,   85,  129,

%e 4   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,

%e 5   | 1,  3,  9, 17,  19,  25,  33,  35,  49,  51,   65,   67,   73,   97,

%e 6   | 1,  5,  9, 17,  21,  33,  37,  41,  65,  69,   73,   81,   85,  129,

%e 7   | 1,  3,  7,  9,  15,  17,  31,  33,  51,  63,   65,   73,   99,  103,

%e 8   | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,

%e 9   | 1,  3,  5,  7,  17,  21,  33,  35,  49,  65,   67,   69,   71,   81,

%e 10  | 1,  3,  9, 17,  19,  25,  33,  35,  49,  51,   65,   67,   73,   97,

%e 11  | 1,  7, 15, 17,  31,  33,  63,  65, 127, 129,  255,  257,  273,  455,

%e 12  | 1,  5,  9, 17,  21,  33,  37,  41,  65,  69,   73,   81,   85,  129,

%e 13  | 1,  7, 15, 17,  31,  33,  63,  65, 127, 129,  255,  257,  273,  455,

%e 14  | 1,  3,  7,  9,  15,  17,  31,  33,  51,  63,   65,   73,   99,  103,

%e 15  | 1, 17, 33, 65, 129, 257, 273, 513, 529, 545, 1025, 1041, 1057, 1089,

%e 16  | 1,  3,  5,  7,   9,  11,  13,  15,  17,  19,   21,   23,   25,   27,

%e 17  | 1,  3,  5,  7,   9,  11,  13,  15,  33,  37,   41,   45,   65,   67,

%e 18  | 1,  3,  5,  7,  17,  21,  33,  35,  49,  65,   67,   69,   71,   81,

%e 19  | 1,  5, 15, 31,  33,  63,  65, 127, 129, 133,  161,  255,  257,  261,

%e 20  | 1,  3,  9, 17,  19,  25,  33,  35,  49,  51,   65,   67,   73,   97,

%e 21  | 1,  3,  5,  9,  15,  21,  31,  33,  45,  47,   61,   63,   65,   67,

%o (PARI)

%o up_to = 105;

%o A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);

%o A391735_sq(n,k) = forstep(i=1,oo,2, if(!(A048720(n,i)%i), if(k>1, k--, return(i))));

%o 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); };

%o v391735 = A391735list(up_to);

%o A391735(n) = v391735[n];

%Y Cf. A048720, A391736.

%Y Row 1: A005408.

%Y Column 2: A391572.

%Y Cf. also A391725.

%K nonn,base,tabl

%O 1,2

%A _Antti Karttunen_, Dec 18 2025