A325560 a(n) is the number of divisors d of n such that A048720(d,k) = n for some k.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 3, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 3, 4, 3, 9, 2, 4, 3, 8, 2, 6, 2, 6, 6, 4, 2, 10, 3, 4, 4, 6, 2, 8, 2, 8, 3, 4, 2, 12, 2, 4, 6, 7, 3, 6, 2, 6, 2, 6, 2, 12, 2, 4, 5, 6, 2, 6, 2, 10, 2, 4, 2, 9, 4, 4, 2, 8, 2, 12, 2, 6, 3, 4, 4, 12, 2, 6, 4, 6, 2, 8, 2, 8, 5

Offset

1,2

Comments

a(n) is the number of divisors d of n such that when the binary expansion of d is converted to a (0,1)-polynomial (e.g., 13=1101[2] encodes X^3 + X^2 + 1), that polynomial is a divisor of the (0,1)-polynomial similarly converted from n, when the polynomial division is done over field GF(2).

Links

Antti Karttunen, Table of n, a(n) 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]

Formula

For all n, A325565(n) <= a(n) <= min(A000005(n), A091220(n)).

Example

39 = 3*13 has four divisors 1, 3, 13, 39, of which all other divisors except 13 are counted because we have A048720(1,39) = A048720(39,1) = A048720(3,29) = 39, but A048720(13,u) is not equal to 39 for any u, thus a(39) = 3. See also the example in A325563.

Prog

(PARI) A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n, d, my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };

Crossrefs

Cf. A000005, A048720, A091220, A325559 (positions of 2's), A325563, A325565.

Sequence in context: A365173 A366991 A365680 * A318412 A365208 A322986

Adjacent sequences: A325557 A325558 A325559 * A325561 A325562 A325563

Keyword

nonn

Author

Antti Karttunen, May 11 2019

Status

approved