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A062799
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Inverse Moebius transform of A001221, the number of distinct prime factors of n.
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9
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0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 12, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 17, 1, 4, 7, 7, 4, 12, 1, 13, 4, 4, 1, 20, 4, 4, 4, 10, 1, 20, 4, 7, 4, 4, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Let us say that two divisors d_1 and d_2 of n are adjacent divisors if d_1/d_2 or d_2/d_1 is a prime. Then a(n) is the number of all pairs of adjacent divisors of n. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 16 2010]
Equivalent to the preceding comment: a(n) is the number of edges in the directed multigraph on tau(n) vertices, vertices labelled by the divisors d_i of n, where edges connect vertex(d_i) and vertex(d_j) if the ratio of the labels is a prime. - R. J. Mathar, Sep 23 2011
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LINKS
| Psychedelic Geometry Blogspot, CURIOUS SERIES-002 [From Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 08 2009]
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FORMULA
| a(n)=Sum{A001221[d]}, where d runs over divisors of n.
Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 08 2009: (Start)
a(s)=omega(s)*2^(omega(s)-1), if s is squarefree (A005117) where omega is A001221
a(n)<=(omega(n)*tau_2(n))-1, where tau_2(n) is A000005 and n>1 (End)
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EXAMPLE
| n = 255: divisors = {1, 3, 5, 15, 17, 51, 85, 255}, a(255) = 0+1+1+2+1+2++2+3 = 12
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MAPLE
| read("transforms") ;
A001221 := proc(n)
nops(numtheory[factorset](n)) ;
end proc:
omega := [seq(A001221(n), n=1..80)] ;
ones := [seq(1, n=1..80)] ;
DIRICHLET(ones, omega) ; # R. J. Mathar, Sep 23 2011
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MATHEMATICA
| f[n_] := Block[{d = Divisors[n], c = l = 0, k = 2}, l = Length[d]; While[k < l + 1, c = c + Length[ FactorInteger[ d[[k]] ]]; k++ ]; Return[c]]; Table[f[n], {n, 1, 100} ]
Contribution from Barbarel Tres Mil (barbarel3000(AT)yahoo.es), Sep 08 2009: (Start)
omega[n_] := Length[FactorInteger[n]]; SetAttributes[omega, Listable]
omega[1] := 0
A062799[n_] := Plus @@ omega[Divisors[n]] (End)
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CROSSREFS
| Cf. A001221.
Sequence in context: A101261 A067614 A113901 * A063647 A077808 A021471
Adjacent sequences: A062796 A062797 A062798 * A062800 A062801 A062802
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Jul 19 2001
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