cp's OEIS Frontend

Also a(n)-1 is the number of 1's in the truth table for the lexicographically least de Bruijn cycle (Fredricksen).

In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system. - Paul Cantrell, Dec 28 2011

Also, minimum cardinality of an unavoidable set of length-n binary words (Champarnaud, Hansel, Perrin). - Jeffrey Shallit, Jan 10 2019

(1/n) * Dirichlet convolution of phi(n) and 2^n, n>0. - Richard L. Ollerton, May 06 2021

From Jianing Song, Nov 13 2021: (Start)

a(n) is even for n != 0, 2. Proof: write n = 2^e * s with odd s, then a(n) * s = Sum_{d|s} Sum_{k=0..e} phi((2^e*s)/(2^k*d)) * 2^(2^k*d-e) = Sum_{d|s} Sum_{k=0..e-1} phi(s/d) * 2^(2^k*d-k-1) + Sum_{d|s} phi(s/d) * 2^(2^e*d-e) == Sum_{k=0..e-1} 2^(2^k*s-k-1) + 2^(2^e*s-e) == Sum_{k=0..min{e-1,1}} 2^(2^k*s-k-1) (mod 2). a(n) is odd if and only if s = 1 and e-1 = 0, or n = 2.

a(n) == 2 (mod 4) if and only if n = 1, 4 or n = 2*p^e with prime p == 3 (mod 4).

a(n) == 4 (mod 8) if and only if n = 2^e, 3*2^e for e >= 3, or n = p^e, 4*p^e != 12 with prime p == 3 (mod 4), or n = 2s where s is an odd number such that phi(s) == 4 (mod 8). (End)

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.

Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).

Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015

Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016

From Gus Wiseman, Sep 14 2019: (Start)

Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:

1 2 3 4 5 6 7 8

123 234 135 246 147 258

345 456 357 468

12345 1236 567 678

1456 2347 1348

23456 2567 1568

12467 3458

13457 3678

34567 12458

1234567 14578

23578

24568

45678

123468

135678

2345678

(End)

Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

From Gus Wiseman, Sep 14 2019: (Start)

Also the number of subsets of {1..n} that are empty or contain n and have integer mean. If the subsets are not required to contain n, we get A327475. For example, the a(1) = 2 through a(6) = 12 subsets are:

{} {} {} {} {} {}

{1} {2} {3} {4} {5} {6}

{1,3} {2,4} {1,5} {2,6}

{1,2,3} {2,3,4} {3,5} {4,6}

{1,3,5} {1,2,6}

{3,4,5} {1,5,6}

{1,2,4,5} {2,4,6}

{1,2,3,4,5} {4,5,6}

{1,2,3,6}

{1,4,5,6}

{2,3,5,6}

{2,3,4,5,6}

(End)

Dirichlet convolution of phi(n) and 2^n. - Richard L. Ollerton, May 06 2021