OFFSET
1,2
COMMENTS
Sum of GCD's of parts in all compositions of n. - Vladeta Jovovic, Aug 13 2003
From Petros Hadjicostas, Dec 07 2017: (Start)
It also equals the sum of all lengths of all cyclic compositions of n. This was proved in Perez (2008).
The bivariate g.f. for the number b(n,k) of all cyclic of compositions of n with k parts is Sum_{n,k>=1} b(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(1 - y^s*Sum_{t>=1} x^{s*t}) = -Sum_{s>=1} (phi(s)/s)*log(1 - y^s*x^s/(1-x^s)). See, for example, Hadjicostas (2016). Differentiating w.r.t. y and setting y = 1, we get Sum_{n>=1} a(n)*x^n = Sum_{n>=1} (Sum_{k=1..n} b(n,k)*k)*x^n = Sum_{s>=1} phi(s)*x^s/(1-2*x^s).
(End)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..3322
P. Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, J. Integer Sequences 19 (2016), Article 16.8.2.
R. A. Perez, Compositions versus cyclic compositions, JP Journal of Algebra, Number Theory and Applications, Vol. 12, Issue 1 (2008), pp. 41-48.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See pp. 47-48, 50.
FORMULA
a(n) = A053635(n)/2.
a(n) = (1/2)* Sum_{d|n} phi(d)*2^(n/d), n >= 1.
G.f.: Sum_{s>=1} phi(s)*x^s/(1-2*x^s). - Petros Hadjicostas, Dec 07 2017
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Feb 07 2019
a(n) = Sum_{k=1..n} 2^(gcd(k, n) - 1). - Seiichi Manyama, Apr 17 2021
a(n) = Sum_{k=1..n} 2^(n/gcd(n,k) - 1)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021
EXAMPLE
For the compositions of n=4 we have a(4) = gcd(4) + gcd(1,3) + gcd(3,1) + gcd(2,2) + gcd(2,1,1) + gcd(1,2,1) + gcd(1,1,2) + gcd(1,1,1,1) = 4 + 1 + 1 + 2 + 1 + 1 + 1 + 1 = 12. Also, for cyclic compositions of n=4, we have length(4) + length(1,3) + length(2,2) + length(1,1,2) + length(1,1,1,1) = 1 + 2 + 2 + 3 + 4 = 12.
MATHEMATICA
Table[Sum[EulerPhi[d]*2^(n/d-1), {d, Divisors[n]}], {n, 1, 40}] (* Vaclav Kotesovec, Feb 07 2019 *)
PROG
(PARI) a(n) = sum(k=1, n, 2^(gcd(k, n)-1)); \\ Seiichi Manyama, Apr 17 2021
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*2^(d-1)); \\ Seiichi Manyama, Apr 17 2021
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-2*x^k))) \\ Seiichi Manyama, Apr 17 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved