A085945 Number of subsets of {1,2,...,n} with relatively prime elements.

1, 2, 5, 11, 26, 53, 116, 236, 488, 983, 2006, 4016, 8111, 16238, 32603, 65243, 130778, 261566, 523709, 1047479, 2095988, 4192115, 8386418, 16772858, 33550058, 67100393, 134209001, 268418531, 536853986, 1073707991, 2147449814, 4294900694, 8589866963

Offset: 1

Author

Vladeta Jovovic, Aug 17 2003

Keywords

• nonn

Examples

			For n=4 there are 11 such subsets: {1}, {1,2}, {1,3}, {1,4}, {2,3}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3321 (first 300 terms from T. D. Noe)
• Mohamed Ayad and Omar Kihel, The number of relatively prime subsets of {1,2,...,n}, Integers 9, No. 2, Article A14, 163-166 (2009).
• M. Ayad, V. Coia and O. Kihel, The Number of Relatively Prime Subsets of a Finite Union of Sets of Consecutive Integers, J. Int. Seq. 17 (2014) # 14.3.7, Theorem 1.
• Mohamed El Bachraoui and Florian Luca, On a Diophantine equation of Ayad and Kihel, Quaestiones Mathematicae, Volume 35, Issue 2, pages 235-243, 2012; DOI:10.2989/16073606.2012.697265. - _N. J. A. Sloane_, Nov 29 2012
• Adrian Łydka, On some properties of the function of the number of relatively prime subsets of {1,2,...,n}, arXiv:1910.02418 [math.NT], 2019.
• Melvyn B. Nathanson, Affine Invariants, Relatively Prime Sets and a Phi Function for Subsets of {1, 2, ..., n}, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A1.
• P. Pongsriiam, Relatively Prime Sets, Divisor Sums, and Partial Sums, arXiv:1306.4891 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.9.1.
• P. Pongsriiam, A remark on relative prime sets, arXiv:1306.2529 [math.NT], 2013.
• P. Pongsriiam, A remark on relative prime sets, Integers 13 (2013), A49.
• M. Tang, Relatively Prime Sets and a Phi Function for Subsets of {1, 2, ... , n}, J. Int. Seq. 13 (2010) # 10.7.6.
• László Tóth, Menon-type identities concerning subsets of the set {1,2,...,n}, arXiv:2109.06541 [math.NT], 2021.

Crossrefs

Cf. A000740, A109511.

Row sums of A143446.

Column k=2 of A143327.

Programs

Maple

b:= n-> add(`if`(d=n, 2^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..35);  # Alois P. Heinz, Oct 05 2018

Mathematica

Table[Sum[MoebiusMu[k] (2^Floor[n/k] - 1), {k, 1, n}], {n, 1, 31}]  (* Geoffrey Critzer, Jan 03 2012 *)

PARI

a(n)=sum(k=1,n,moebius(k)*(2^floor(n/k)-1)) \\ Charles R Greathouse IV, Feb 04 2013

Formula

Partial sums of A000740. G.f.: 1/(1-x)* Sum_{k>0} mu(k)*x^k/(1-2*x^k).

a(n) = 2^n - A109511(n) - 1. - Reinhard Zumkeller, Jul 01 2005

a(n) = Sum_{k=1..n} mu(k)*(2^floor(n/k)-1). - Geoffrey Critzer, Jan 03 2012