A034729 - OEIS

%I #45 Jun 26 2024 19:36:15

%S 1,3,5,11,17,39,65,139,261,531,1025,2095,4097,8259,16405,32907,65537,

%T 131367,262145,524827,1048645,2098179,4194305,8390831,16777233,

%U 33558531,67109125,134225995,268435457,536887863,1073741825,2147516555,4294968325,8590000131

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

%C Dirichlet convolution of b_n=1 with c_n = 2^(n-1).

%C Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - _Gary W. Adamson_, Aug 14 2008

%C 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 T. D. Noe, <a href="/A034729/b034729.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{n>0} x^n/(1-2*x^n). - _Vladeta Jovovic_, Nov 14 2002

%F a(n) = 1/2 * A055895(n). - _Joerg Arndt_, Aug 14 2012

%F G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - _Paul D. Hanna_, Aug 21 2014

%F G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - _Paul D. Hanna_, Aug 21 2014

%F a(n) ~ 2^(n-1). - _Vaclav Kotesovec_, Sep 09 2014

%F a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - _Gus Wiseman_, Sep 16 2018

%F a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - _Gus Wiseman_, Sep 16 2018

%e From _Gus Wiseman_, Sep 16 2018: (Start)

%e The a(4) = 11 constant multiset partitions:

%e   (1)(1)(1)(1)

%e     (11)(11)

%e     (12)(12)

%e      (1111)

%e      (1222)

%e      (1122)

%e      (1112)

%e      (1233)

%e      (1223)

%e      (1123)

%e      (1234)

%e (End)

%p seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # _Robert Israel_, Aug 22 2014

%t Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* _Vaclav Kotesovec_, Sep 08 2014 *)

%o (PARI) A034729(n) = sumdiv(n,k,2^(k-1)) \\ _Michael B. Porter_, Mar 11 2010

%o (PARI) {a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}

%o for(n=1,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 21 2014

%o (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 for(n=1,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 21 2014

%o (Python)

%o from sympy import divisors

%o def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jul 15 2022

%o (Magma)

%o A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;

%o [A034729(n): n in [1..40]]; // _G. C. Greubel_, Jun 26 2024

%o (SageMath)

%o def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))

%o [A034729(n) for n in range(1,41)] # _G. C. Greubel_, Jun 26 2024

%Y Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.

%Y Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.

%Y Cf. A289508.

%Y 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 nonn

%O 1,2

%A _Erich Friedman_