cp's OEIS Frontend

A325559 Numbers n such that for any divisor d of n, and some integer k, A048720(d,k) = n only for trivial cases d=1 and d=n.

%I A325559 #19 May 11 2019 18:32:57
%S A325559 2,3,5,7,11,13,17,19,23,25,29,31,37,41,43,47,53,55,59,61,67,69,71,73,
%T A325559 77,79,81,83,87,89,91,97,101,103,107,109,113,115,117,121,127,131,137,
%U A325559 139,143,145,149,151,157,163,167,169,171,173,179,181,185,191,193,197,199,203,205,209,211,213,223,227,229,233
%N A325559 Numbers n such that for any divisor d of n, and some integer k, A048720(d,k) = n only for trivial cases d=1 and d=n.
%C A325559 These are numbers n such that there are only two divisor pairs (d, n/d) [namely, the trivial pairs (1, n) and (n, 1)] that satisfy the condition that when their binary expansions are converted to (0,1)-polynomials (e.g., 13=1101[2] encodes X^3 + X^2 + 1), then their product is the (0,1)-polynomial similarly converted from n, when the multiplication is done over field GF(2).
%C A325559 Differs from A206074 for the first time at n=173, where a(173) = 555, a value missing from A206074, while the first three terms of A206074 not present in this sequence are k = 689, 781 and 913, for all of which A325560(k) = 3, not 2.
%H A325559 Antti Karttunen, <a href="/A325559/b325559.txt">Table of n, a(n) for n = 1..25000</a>
%H A325559 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>
%o A325559 (PARI)
%o A325559 A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };
%o A325559 isA325559(n) = (2 == A325560(n));
%Y A325559 Cf. A048720, A206074.
%Y A325559 Positions of 2's in A325560, positions of 1's in A325563 (after the initial 1), fixed points of A325643 (after the initial 1).
%Y A325559 Some subsequences: A257688 (after its initial 1), A325386 (the remaining terms).
%K A325559 nonn
%O A325559 1,1
%A A325559 _Antti Karttunen_, May 11 2019