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%I #11 Dec 18 2025 22:24:35
%S 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,
%T 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,
%U 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
%N 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.
%C Array A391735 gives the corresponding odd numbers i that satisfy the requirement.
%C 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.
%H Antti Karttunen, <a href="/A391736/b391736.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) = 2*A(n, k).
%F For all n, k: A391736(n,k) * A391735(n,k) = A048720(n, A391735(n,k)).
%F For all n, k: A(n, k) <= n. [Implied by above, as x*y >= A048720(x,y)]
%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 15 16 17
%e ----+--------------------------------------------------------------------
%e 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
%e 2 | 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
%e 3 | 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
%e 4 | 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
%e 5 | 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
%e 6 | 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
%e 7 | 7, 3, 3, 7, 3, 7, 3, 7, 3, 3, 7, 7, 3, 3, 3, 3, 7,
%e 8 | 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,
%e 9 | 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9,
%e 10 | 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,
%e 11 | 11, 7, 7, 11, 7, 11, 7, 11, 7, 11, 7, 11, 11, 7, 7, 11, 11,
%e 12 | 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12,
%e 13 | 13, 5, 5, 13, 5, 13, 5, 13, 5, 13, 5, 13, 13, 5, 5, 13, 13,
%e 14 | 14, 6, 6, 14, 6, 14, 6, 14, 6, 6, 14, 14, 6, 6, 6, 6, 14,
%e 15 | 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15,
%e 16 | 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
%e 17 | 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17,
%e 18 | 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18,
%e 19 | 19, 19, 15, 15, 19, 15, 19, 15, 19, 19, 19, 15, 19, 19, 19, 15, 19,
%e 20 | 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20,
%e 21 | 21, 21, 13, 21, 13, 13, 13, 21, 13, 13, 13, 13, 21, 21, 21, 13, 13,
%o (PARI)
%o up_to = 105;
%o A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
%o A391736_sq(n,k) = forstep(i=1,oo,2, if(!(A048720(n,i)%i), if(k>1, k--, return(A048720(n,i)/i))));
%o 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); };
%o v391736 = A391736list(up_to);
%o A391736(n) = v391736[n];
%Y Cf. A115861, A391735.
%Y Row 1: A000012.
%Y Column 2: A391573.
%Y Cf. also A391726.
%K nonn,base,tabl
%O 1,3
%A _Antti Karttunen_, Dec 18 2025