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A069513
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Characteristic function of the prime powers p^k, k >= 1.
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28
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0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
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OFFSET
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1,1
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COMMENTS
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Also, number of abelian indecomposable groups of order n. - Kevin Lamoreau, Mar 13 2023
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LINKS
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FORMULA
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a(n) = Sum_{d|n} bigomega(d)*mu(n/d); equivalently, Sum_{d|n} a(d) = bigomega(n); equivalently, Möbius transform of bigomega(n).
Dirichlet g.f.: ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s - 1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
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MAPLE
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if n = 1 then
0 ;
0;
else
1 ;
end if ;
end proc:
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MATHEMATICA
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PROG
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(PARI) for(n=1, 120, print1(omega(n)==1, ", "))
(Haskell)
a069513 1 = 0
(Python)
from sympy import primefactors
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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Moved original definition to formula line. Used comment (that I previously added) as definition. - Daniel Forgues, Mar 08 2009
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STATUS
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approved
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