A391571 - OEIS

%I #30 Dec 19 2025 12:40:46

%S 1,2,7,4,5,14,11,8,9,10,31,28,29,22,27,16,17,18,23,20,21,62,59,56,57,

%T 58,43,44,37,54,35,32,33,34,39,36,37,46,63,40,41,42,127,124,125,118,

%U 123,112,113,114,119,116,117,86,99,88,89,74,111,108,109,70,107,64,65,66,71,68,69,78,75,72,73,74,95,92,93,126,91

%N a(n) is the value of A280500(n*k, k) for the smallest odd k > 1 for which this value is not zero, where A280500 implements the carryless base-2 division, and returns 0 if the division leaves nonzero remainder.

%C Search for the smallest odd k > 1 such that k*n = A048720(k,m) for some m, and then set a(n) = m. A391570(n) gives the k.

%C The smallest term k of A004780 for which a(k) = k is A004780(940) = 1117. See A391581.

%H Antti Karttunen, <a href="/A391571/b391571.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 A391570(n) * n = A048720(A391570(n), a(n)).

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

%F a(A003714(n)) = A003714(n), and a(A391581(n)) = A391581(n).

%o (PARI) A391571(n) = forstep(k=3,oo,2, my(Pnk=Pol(binary(n*k))*Mod(1, 2), Pk=Pol(binary(k))*Mod(1, 2)); if(0==lift(Pnk % Pk), return(fromdigits(Vec(lift(Pnk / Pk)),2))));

%Y Cf. A048720, A280500, A391570, A391580 (terms k for which a(k) > k).

%Y Column 2 of A391726 (cf. also A391583, which is column 3. Compare also their scatter plots).

%Y Fixed points: (A003714 U A391581).

%Y Cf. also A115857, A391572, A391573.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Dec 15 2025