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A072873 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) ). 15
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; text; internal format)
OFFSET

1,2

COMMENTS

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

Multiplicative closure of A051674. - Reinhard Zumkeller, Jan 21 2012

Where records occur in A085731. - Paolo P. Lava, Oct 02 2012

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

REFERENCES

See A003415.

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Nathaniel Johnston)

FORMULA

A124010(a(n),k) mod A027748(a(n),k) = 0 for k = 1 .. A001221(a(n)). - Reinhard Zumkeller, Jan 21 2012

Sum_{n>=1} 1/a(n) = Product_{p prime} p^p/(p^p-1) = 1.38506028520448917638... - Amiram Eldar, Sep 27 2020

EXAMPLE

108 is in the sequence because 108 = 2^2*3^3 and 2/2 + 3/3 = 2 is an integer.

MAPLE

with(numtheory);

A072873:= proc(i)

local a, n, p;

for n from 1 to i do

  a:=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);

  if trunc(a/n)=a/n then print(n); fi;

od; end:

A072873(100000000); # Paolo P. Lava, Oct 18 2012

MATHEMATICA

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 *)

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

(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

CROSSREFS

Cf. A003415, A051674, A027748, A085731, A048102, A124010.

Sequence in context: A328415 A097764 A227993 * A072653 A008478 A201009

Adjacent sequences:  A072870 A072871 A072872 * A072874 A072875 A072876

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jul 28 2002

EXTENSIONS

More terms from T. D. Noe, Jan 04 2006

STATUS

approved

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Last modified March 8 13:25 EST 2021. Contains 341948 sequences. (Running on oeis4.)