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A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

%I A034729 #45 Jun 26 2024 19:36:15
%S A034729 1,3,5,11,17,39,65,139,261,531,1025,2095,4097,8259,16405,32907,65537,
%T A034729 131367,262145,524827,1048645,2098179,4194305,8390831,16777233,
%U A034729 33558531,67109125,134225995,268435457,536887863,1073741825,2147516555,4294968325,8590000131
%N A034729 a(n) = Sum_{ k, k|n } 2^(k-1).
%C A034729 Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
%C A034729 Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - _Gary W. Adamson_, Aug 14 2008
%C A034729 Number of constant multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - _Gus Wiseman_, Sep 16 2018
%H A034729 T. D. Noe, <a href="/A034729/b034729.txt">Table of n, a(n) for n = 1..1000</a>
%F A034729 G.f.: Sum_{n>0} x^n/(1-2*x^n). - _Vladeta Jovovic_, Nov 14 2002
%F A034729 a(n) = 1/2 * A055895(n). - _Joerg Arndt_, Aug 14 2012
%F A034729 G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - _Paul D. Hanna_, Aug 21 2014
%F A034729 G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - _Paul D. Hanna_, Aug 21 2014
%F A034729 a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Sep 09 2014
%F A034729 a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - _Gus Wiseman_, Sep 16 2018
%F A034729 a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - _Gus Wiseman_, Sep 16 2018
%e A034729 From _Gus Wiseman_, Sep 16 2018: (Start)
%e A034729 The a(4) = 11 constant multiset partitions:
%e A034729   (1)(1)(1)(1)
%e A034729     (11)(11)
%e A034729     (12)(12)
%e A034729      (1111)
%e A034729      (1222)
%e A034729      (1122)
%e A034729      (1112)
%e A034729      (1233)
%e A034729      (1223)
%e A034729      (1123)
%e A034729      (1234)
%e A034729 (End)
%p A034729 seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # _Robert Israel_, Aug 22 2014
%t A034729 Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* _Vaclav Kotesovec_, Sep 08 2014 *)
%o A034729 (PARI) A034729(n) = sumdiv(n,k,2^(k-1)) \\ _Michael B. Porter_, Mar 11 2010
%o A034729 (PARI) {a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}
%o A034729 for(n=1,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 21 2014
%o A034729 (PARI) {a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,1/(1 - x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
%o A034729 for(n=1,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 21 2014
%o A034729 (Python)
%o A034729 from sympy import divisors
%o A034729 def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jul 15 2022
%o A034729 (Magma)
%o A034729 A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
%o A034729 [A034729(n): n in [1..40]]; // _G. C. Greubel_, Jun 26 2024
%o A034729 (SageMath)
%o A034729 def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
%o A034729 [A034729(n) for n in range(1,41)] # _G. C. Greubel_, Jun 26 2024
%Y A034729 Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.
%Y A034729 Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.
%Y A034729 Cf. A289508.
%Y A034729 Sums of the form Sum_{d|n} q^(d-1): this sequence (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).
%K A034729 nonn
%O A034729 1,2
%A A034729 _Erich Friedman_