A365410 Parity of A000070.

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1

Offset

0

Comments

Parity of the number of 1's in all partitions of (n + 1).

Hence parity of the number of 1's in all divisors of the terms in the (n + 1)st row of the triangle A176206.

Links

Table of n, a(n) for n=0..105.

Formula

a(n) = A000035(A000070(n)).

Example

For n = 5 the partitions of 6 and the divisors of terms of the 6th row of A176206 show an example of the correspondence divisor/part in an arrangement as shown below:
.
    6
    3, 3
    4, 2
    2, 2, 2
    5, 1
    3, 2, 1
    4, 1, 1
    2, 2, 1, 1
    3, 1, 1, 1
    2, 1, 1, 1, 1
    1, 1, 1, 1, 1, 1
   ------------------
    1  2  3        6
    1           5
    1  2     4
    1  2     4
    1     3
    1     3
    1     3
    1  2
    1  2
    1  2
    1  2
    1  2
    1
    1
    1
    1
    1
    1
    1
.
In the above arrangement appear the partitions of 6 and below the divisors of [6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1] the 6th row of A176206.
The parts of all partitions of 6 are also the divisors of the terms of the 6th row of A176206 hence the number of 1's in all partitions of 6 equals the number of 1's in all divisors of the terms in the 6th row of A176206. In each case there are nineteen 1's. The parity of 19 is 1 so a(5) = 1.

Mathematica

Mod[Accumulate@ Array[PartitionsP, 120, 0], 2] (* Michael De Vlieger, Oct 01 2023 *)

Crossrefs

Cf. A000035, A000041, A000070, A040051, A176206, A336811, A338156.

Sequence in context: A174889 A005171 A283265 * A181406 A285252 A076404

Adjacent sequences: A365407 A365408 A365409 * A365411 A365412 A365413

Keyword

nonn

Author

Omar E. Pol, Sep 25 2023

Status

approved