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

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, 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, 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

Offset: 1

Author

Antti Karttunen, May 09 2019

Keywords

• nonn

Comments

Equally, a(n) is number of such pairs of natural numbers t, u that A048720(t,u) = n and A065620(t)*u = n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences defined by congruent products between domains N and GF(2)[X]
• Index entries for sequences related to polynomials in ring GF(2)[X]

Crossrefs

Cf. A000005, A000265, A001511, A048720, A065620, A065621, A091220, A115872, A325566, A325567, A325568, A325569, A325570 (positions of ones).

Programs

PARI

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
A325565(n) = sumdiv(n,d,A048720(A065621(d),n/d)==n);

PARI

A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<A065620
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))))); };

Formula

a(n) = Sum_{d|n} [A048720(A065621(d),n/d) == n], where [ ] is the Iverson bracket.

a(n) / a(A000265(n)) = A001511(n).

a(n) <= A000005(n) for all n.

a(n) <= A091220(n) for all n.