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A034729
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a(n) = Sum_{ k, k|n } 2^(k-1).
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44
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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
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OFFSET
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1,2
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COMMENTS
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Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
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
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LINKS
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FORMULA
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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
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EXAMPLE
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The a(4) = 11 constant multiset partitions:
(1)(1)(1)(1)
(11)(11)
(12)(12)
(1111)
(1222)
(1122)
(1112)
(1233)
(1223)
(1123)
(1234)
(End)
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MAPLE
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seq(add(2^(k-1), k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014
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MATHEMATICA
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Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k), {k, 1, 30}], {x, 0, 30}], x]] (* Vaclav Kotesovec, Sep 08 2014 *)
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PROG
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(PARI) {a(n)=polcoeff(sum(m=1, n, 2^(m-1)*x^m/(1-x^m +x*O(x^n))), n)}
(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)}
(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
(Magma)
A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
(SageMath)
def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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