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

1,2

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

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

Sequence in context: A038577 A028925 A028924 * A054064 A246866 A053479

Adjacent sequences: A034735 A034736 A034737 * A034739 A034740 A034741

Keyword

nonn

Author

Erich Friedman

Status

approved