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%I #27 Dec 23 2025 15:27:45
%S 1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,6,5,4,3,2,1,7,6,5,4,3,2,1,8,7,6,5,4,3,
%T 2,1,9,8,7,6,6,4,4,2,1,10,9,8,7,7,5,7,3,2,1,11,10,9,8,8,6,8,4,3,2,1,
%U 12,11,10,9,9,7,9,5,4,3,2,1,13,12,12,10,12,8,14,6,5,4,3,2,1,14,13,14,11,14,9,15,7,6,6,4,3,2,1
%N Square array read by descending antidiagonals: A(n, k) is the k-th natural number i that satisfies i*n = A048720(i,m) for some m, where A048720 is carryless base-2 multiplication.
%C Array A391926 gives the corresponding m.
%H Antti Karttunen, <a href="/A391925/b391925.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 For all n, k: n * A(n,k) = A048720(A(n,k), A391926(n,k)).
%e Note: The array does not list the initial 0 of each row, which however is included in the row sequences given at right margin. After the semicolon are listed the subsequences of that row which satisfy the relationship for a particular m (given in parentheses, and followed by ? if it is so far only a conjecture). Array A115872 lists the subsequences with m = A065621(n), where n is the row number.
%e The top left corner of the array:
%e n\k | 1 2 3 4 5 6 7 8 9 10 11
%e ----+---------------------------------------
%e 1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, A001477
%e 2 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, (ditto for all rows 2^e)
%e 3 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, A391585; A003714 (3), A048717 (7)
%e 4 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
%e 5 | 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, A391740; A048716 (5), A115770 (13, ?)
%e 6 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12,
%e 7 | 1, 2, 4, 7, 8, 9, 14, 15, 16, 17, 18, A391742; A048715 (7), A115770 (11)
%e 8 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
%e 9 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, A391744; A115845 (9), A115801 (25)
%e 10 | 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15,
%e 11 | 1, 2, 3, 4, 6, 8, 12, 15, 16, 17, 24, A391846; A048718 (11), A115803 (31)
%e 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12,
%e 13 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, A391848; A048718 (13), A115772 (21), A115805 (29)
%e 14 | 1, 2, 4, 7, 8, 9, 14, 15, 16, 17, 18,
%e 15 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, A391850; A048718 (15), A115801 (19, ?), A115774 (23), ...
%e 16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and A115807 (27)
%e 17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, A391852; A115847 (17), A115809 (49)
%e 18 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14,
%e 19 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, A391854; A115805 (23, ?), A115874 (55)
%e 20 | 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15,
%e 21 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, A391856; A115422 (21), A115774 (29, ?), A115809 (61, ?)
%e 22 | 1, 2, 3, 4, 6, 8, 12, 15, 16, 17, 24,
%e 23 | 1, 2, 4, 7, 8, 14, 16, 28, 31, 32, 33,
%e 24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12,
%e 25 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, A391858; A391737 (25), A391738 (41), A391739 (57)
%o (PARI)
%o up_to = 105;
%o A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
%o A391925_sq(n,k) = for(i=1,oo, 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(i))));
%o A391925list(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] = A391925_sq(col,(a-(col-1))))); (v); };
%o v391925 = A391925list(up_to);
%o A391925(n) = v391925[n];
%Y Cf. A048720, A065621, A391926.
%Y Column 1: A000012.
%Y Row 1: A001477 (also occurs as every 2^e:th row).
%Y Other rows: A391585 (row 3), A391740 (row 5), A391742 (row 7), A391744 (row 9), A391846 (row 11), A391848 (row 13), A391850 (row 15), A391852 (row 17), A391854 (row 19), A391856 (row 21), A391858 (row 25), A391860 (row 49).
%Y Cf. A115872 (subarray whose rows are subsequences of these rows), A391725 (subarray with rows listing only the odd terms).
%K nonn,base,tabl
%O 1,2
%A _Antti Karttunen_, Dec 23 2025