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A072873
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Numbers n such that sum( e(i)/p(i) ) is an integer, where the prime factorization of n is prod ( p(i)^e(i) ).
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9
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1, 4, 16, 27, 64, 108, 256, 432, 729, 1024, 1728, 2916, 3125, 4096, 6912, 11664, 12500, 16384, 19683, 27648, 46656, 50000, 65536, 78732, 84375, 110592, 186624, 200000, 262144, 314928, 337500, 442368, 531441, 746496, 800000, 823543
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also, numbers n such that n divides n', the arithmetic derivative of n. As shown by Ufnarovski and Ahlander, all terms in this sequence have the form prod_{k=1..r} (pk^pk)^ek, where the pk are primes. The quotient n'/n equals sum_{k=1..r} ek. - T. D. Noe (noe(AT)sspectra.com), Jan 04 2006
Multipliative closure of A051674. [Reinhard Zumkeller, Jan 21 2012]
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REFERENCES
| See A003415.
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LINKS
| Nathaniel Johnston, Table of n, a(n) for n = 1..2500
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FORMULA
| A124010(a(n),k) mod A027748(a(n),k) = 0 for k = 1 .. A001221(a(n)). [Reinhard Zumkeller, Jan 21 2012]
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EXAMPLE
| 108 is in the sequence because 108 = 2^2*3^3 and 2/2+3/3 = 2 is an integer.
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MAPLE
| P:=proc(p) local a, b, m, n, i, ok, t1, t2, t3; a:=0; for n from 1 by 1 to p do b:=1000000000039; ok:=0; if n<=1 then a:=0; ok:=1; fi; if isprime(n) then a:=1; ok:=1; fi; if ok=0 then t1:=ifactor(b*n); m:=nops(t1); t2:=0; for i from 1 to m do t3:=op(i, t1); if nops(t3)=1 then t2:=t2+1/op(t3); else t2:=t2+op(2, t3)/op(op(1, t3)); fi; od; t2:=t2-1/b; a:=n*t2; fi; if (a mod n)=0 then print(n); fi; od; end: P(1000000); [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 13 2009]
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PROG
| (Haskell)
import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null)
a072873 n = a072873_list !! (n-1)
a072873_list = 1 : h empty [1] a051674_list where
h s mcs xs'@(x:xs)
| Set.null s || x < m = h (s `union` fromList (map (* x) mcs)) mcs xs
| otherwise = m : h (s' `union` fromList (map (* m) $ init (m:mcs)))
(m:mcs) xs'
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jan 21 2012
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CROSSREFS
| Cf. A048102.
Sequence in context: A046346 A134330 A097764 * A072653 A008478 A111260
Adjacent sequences: A072870 A072871 A072872 * A072874 A072875 A072876
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KEYWORD
| nonn
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 28 2002
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EXTENSIONS
| More terms from T. D. Noe (noe(AT)sspectra.com), Jan 04 2006
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