A034738 Dirichlet convolution of b_n = 2^(n-1) with phi(n).

1, 3, 6, 12, 20, 42, 70, 144, 270, 540, 1034, 2112, 4108, 8274, 16440, 32928, 65552, 131418, 262162, 524880, 1048740, 2098206, 4194326, 8391024, 16777300, 33558564, 67109418, 134226120, 268435484, 536888520, 1073741854, 2147516736

Offset: 1

Author

Erich Friedman

Keywords

• nonn

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)

Examples

			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.
		

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.

Crossrefs

Cf. A000010, A000740, A034754, A053635, A078392.

Programs

Mathematica

Table[Sum[EulerPhi[d]*2^(n/d-1), {d, Divisors[n]}], {n, 1, 40}] (* Vaclav Kotesovec, Feb 07 2019 *)

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

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