A391573 - OEIS

%I #25 Dec 19 2025 12:40:53

%S 1,2,3,4,5,6,3,8,9,10,7,12,5,6,15,16,17,18,19,20,21,14,19,24,25,10,15,

%T 12,13,30,11,32,33,34,35,36,37,38,35,40,41,42,27,28,45,38,27,48,49,50,

%U 51,20,45,30,47,24,25,26,43,60,21,22,39,64,65,66,67,68,69,70,67,72,73,74,71,76,69,70,63,80,81,82,63

%N a(n) is the value of quotient A048720(n,k)/k for the smallest odd k > 1 for which A048720(n,k) is a multiple of k (in ordinary arithmetic), where A048720 is carryless base-2 multiplication.

%H Antti Karttunen, <a href="/A391573/b391573.txt">Table of n, a(n) for n = 1..16384</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(n) * A391572(n) = A048720(n, A391572(n)).

%F a(n) <= n. [Implied by above, as x*y >= A048720(x,y)]

%F a(A003714(n)) = A003714(n). [A003714 is a proper subset of fixed points, A391579]

%o (PARI)

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

%o A391573(n) = { my(t); forstep(k=3,oo,2, if(!((t=A048720(n,k))%k), return(t/k))); };

%Y Cf. A003714, A048720, A391569 (= n XOR a(n)), A391572, A391578 (k such that a(k) < k), A391579 (fixed points).

%Y Column 2 of A391736.

%Y Cf. also A391570, A391571.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Dec 16 2025