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A325565 a(n) is the number of such divisors d of n that A048720(A065621(d),n/d) is equal to n.

%I A325565 #19 May 11 2019 02:23:38
%S A325565 1,2,1,3,1,2,1,4,2,2,1,3,1,2,1,5,1,4,1,3,2,2,1,4,1,2,1,3,1,2,1,6,2,2,
%T A325565 2,6,1,2,1,4,1,4,1,3,2,2,1,5,2,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,2,4,1,3,
%U A325565 1,4,1,8,1,2,2,3,1,2,1,5,1,2,1,6,1,2,1,4,1,4,1,3,2,2,1,6,1,4,1,3,1,2,1,4,2
%N A325565 a(n) is the number of such divisors d of n that A048720(A065621(d),n/d) is equal to n.
%C A325565 Equally, a(n) is number of such pairs of natural numbers t, u that A048720(t,u) = n and A065620(t)*u = n.
%H A325565 Antti Karttunen, <a href="/A325565/b325565.txt">Table of n, a(n) for n = 1..16384</a>
%H A325565 Antti Karttunen, <a href="/A325565/a325565.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A325565 <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>
%H A325565 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>
%F A325565 a(n) = Sum_{d|n} [A048720(A065621(d),n/d) == n], where [ ] is the Iverson bracket.
%F A325565 a(n) / a(A000265(n)) = A001511(n).
%F A325565 a(n) <= A000005(n) for all n.
%F A325565 a(n) <= A091220(n) for all n.
%o A325565 (PARI)
%o A325565 A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
%o A325565 A065621(n) = bitxor(n-1,n+n-1);
%o A325565 A325565(n) = sumdiv(n,d,A048720(A065621(d),n/d)==n);
%o A325565 (PARI)
%o A325565 A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<<i)); \\ From A065620
%o A325565 A325565(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sum(d=1,n,my(q = Pol(binary(d))*Mod(1, 2)); (0==(p%q) && (n==(A065620(d)*fromdigits(Vec(lift(p/q)),2))))); };
%Y A325565 Cf. A000005, A000265, A001511, A048720, A065620, A065621, A091220, A115872, A325566, A325567, A325568, A325569, A325570 (positions of ones).
%K A325565 nonn
%O A325565 1,2
%A A325565 _Antti Karttunen_, May 09 2019