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 A034729 a(n) = Sum_{ k, k|n } 2^(k-1). 43
 1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145, 524827, 1048645, 2098179, 4194305, 8390831, 16777233, 33558531, 67109125, 134225995, 268435457, 536887863, 1073741825, 2147516555, 4294968325, 8590000131 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Dirichlet convolution of b_n=1 with c_n=2^(n-1). Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - Gary W. Adamson, Aug 14 2008 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 LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 FORMULA G.f.: Sum_{n>0} x^n/(1-2*x^n). - Vladeta Jovovic, Nov 14 2002 a(n) = 1/2 * A055895(n). - Joerg Arndt, Aug 14 2012 G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - Paul D. Hanna, Aug 21 2014 G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - Paul D. Hanna, Aug 21 2014 a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 09 2014 a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - Gus Wiseman, Sep 16 2018 a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - Gus Wiseman, Sep 16 2018 EXAMPLE From Gus Wiseman, Sep 16 2018: (Start) The a(4) = 11 constant multiset partitions: (1)(1)(1)(1) (11)(11) (12)(12) (1111) (1222) (1122) (1112) (1233) (1223) (1123) (1234) (End) MAPLE seq(add(2^(k-1), k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014 MATHEMATICA Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k), {k, 1, 30}], {x, 0, 30}], x]] (* Vaclav Kotesovec, Sep 08 2014 *) PROG (PARI) A034729(n) = sumdiv(n, k, 2^(k-1)) \\ Michael B. Porter, Mar 11 2010 (PARI) {a(n)=polcoeff(sum(m=1, n, 2^(m-1)*x^m/(1-x^m +x*O(x^n))), n)} for(n=1, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 21 2014 (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)} for(n=1, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 21 2014 (Python) from sympy import divisors def A034729(n): return sum(1<<(d-1) for d in divisors(n, generator=True)) # Chai Wah Wu, Jul 15 2022 CROSSREFS Cf. A034730, A051731, A143425, A245282, A248906. Cf. A002033, A003238, A018783, A047968, A052409, A078392. Sequence in context: A155989 A125557 A007455 * A115786 A252089 A128550 Adjacent sequences: A034726 A034727 A034728 * A034730 A034731 A034732 KEYWORD nonn AUTHOR Erich Friedman STATUS approved

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Last modified May 18 15:24 EDT 2024. Contains 372664 sequences. (Running on oeis4.)