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 A034738 Dirichlet convolution of b_n = 2^(n-1) with phi(n). 19
 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 (list; graph; refs; listen; history; text; internal format)
 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 STATUS approved

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)