A295368 For any number n > 0 with s divisors, say d_1, d_2, ..., d_s such that d_1 = 1 < d_2 < ... < d_s = n, the binary representation of a(n) is (d_1 mod 2, d_2 mod 2, ..., d_s mod 2).

1, 2, 3, 4, 3, 10, 3, 8, 7, 10, 3, 40, 3, 10, 15, 16, 3, 42, 3, 36, 15, 10, 3, 160, 7, 10, 15, 36, 3, 178, 3, 32, 15, 10, 15, 328, 3, 10, 15, 144, 3, 170, 3, 36, 63, 10, 3, 640, 7, 42, 15, 36, 3, 170, 15, 144, 15, 10, 3, 2696, 3, 10, 63, 64, 15, 170, 3, 36, 15

Offset: 1

Author

Rémy Sigrist, Feb 18 2018

Keywords

• nonn
• base

Comments

This sequence encodes in binary the parity of the divisors of a number.

For any n > 0, the binary representation of a(n) corresponds to the n-th row of A247795.

For any n > 0, n and a(n) have the same parity.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A000005(n))

Crossrefs

Cf. A000005, A000120, A001227, A016825, A023416, A070939, A183063, A247795.

Programs

Mathematica

Array[FromDigits[Mod[#, 2] & /@ Divisors@ #, 2] &, 69] (* Michael De Vlieger, Feb 18 2018 *)

PARI

a(n) = fromdigits(apply(d -> d%2, divisors(n)), 2)

Formula

a(2^k) = 2^k for any k >= 0.

a(p) = 3 iff p is an odd prime.

a(n) > 3 iff n is composite.

A070939(a(n)) = A000005(n) for any n > 0.

A000120(a(n)) = A001227(n) for any n > 0.

A023416(a(n)) = A183063(n) for any n > 0.

A000120(a(n)) = A023416(a(n)) iff n belongs to A016825.