A072873 Numbers k such that Sum_i ( e(i)/p(i) ) is an integer, where the prime factorization of k is Product_i ( p(i)^e(i) ).

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

Offset: 1

Author

Benoit Cloitre, Jul 28 2002

Keywords

• nonn

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

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

Examples

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

References

• See A003415.

Links

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

Crossrefs

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

Programs

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

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 *)

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()))
A072873_list = list(islice(A072873_gen(),20)) # Chai Wah Wu, Sep 15 2023

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

Extensions

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