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A072873
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Numbers k such that Sum_i ( e(i)/p(i) ) is an integer, where the prime factorization of n is Product_i ( p(i)^e(i) ).
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29
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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
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OFFSET
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1,2
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COMMENTS
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Also, numbers k such that k divides k', the arithmetic derivative of k. As shown by Ufnarovski and Ahlander, all terms in this sequence have the form Product_{j=1..r} (pj^pj)^ej, where the pj are primes. The quotient k'/k equals Sum_{j=1..r} ej. - T. D. Noe, Jan 04 2006
The number of terms < 10^k: 2, 5, 9, 15, 25, 36, 52, 73, 98, 128, 167, 213, 270, 338, 421, 517, 632, 768, 920, 1101, ..., . - Robert G. Wilson v, Jan 19 2016
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REFERENCES
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = Product_{p prime} p^p/(p^p-1) = 1.38506028520448917638... - Amiram Eldar, Sep 27 2020
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EXAMPLE
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108 is in the sequence because 108 = 2^2*3^3 and 2/2 + 3/3 = 2 is an integer.
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MATHEMATICA
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Select[Range[1000000], IntegerQ[Total[#[[2]]/#[[1]]&/@FactorInteger[#]]]&] (* Harvey P. Dale, Jul 04 2014 *)
lst = {}; Do[n = 2^e2*3^e3*5^e5*7^e7; If[n < 10^11, AppendTo[lst, n]], {e2, 0, 36, 2}, {e3, 0, 23, 3}, {e5, 0, 15, 5}, {e7, 0, 13, 7}]; Take[ Sort@ lst, 40] (* Robert G. Wilson v, Jan 19 2016 *)
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PROG
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(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
(PARI) is(n)=my(f=factor(n)); for(i=1, #f~, if(f[i, 2]%f[i, 1], return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2014
(Python)
from itertools import count, islice
from sympy import factorint
def A072873_gen(startvalue=1): # generator of terms >= startvalue
return (k for k in count(max(startvalue, 1)) if not any(e%p for p, e in factorint(k).items()))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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