A201009 Numbers m such that the set of distinct prime divisors of m is equal to the set of distinct prime divisors of the arithmetic derivative m'.

1, 4, 16, 27, 108, 144, 256, 432, 500, 784, 972, 1323, 1728, 2700, 2916, 3125, 3456, 5292, 8788, 11664, 12500, 13068, 15376, 16875, 19683, 20736, 23328, 25000, 27648, 28125, 31212, 34300, 47916, 54000, 57132, 65536, 72000, 78732, 97556, 102400, 103788, 104544

Offset

1,2

Comments

A027748(n,k) = A027748(A003415(n),k) for k=1..A001221(n). - Reinhard Zumkeller, Jan 16 2013

A051674 is a subsequence of this sequence.

Links

Paolo P. Lava and Donovan Johnson, Table of n, a(n) for n = 1..500 (first 100 terms from Paolo P. Lava)

Example

n = 1728 = 2^6*3^3, n' = 6912 = 2^8*3^3 have the same prime factors 2 and 3.

Maple

with(numtheory);
A201009:=proc(q)
local a, b, k, n;
for n from 1 to q do
  a:=ifactors(n)[2]; b:=ifactors(n*add(op(2, p)/op(1, p), p=ifactors(n)[2]))[2];
  if nops(a)=nops(b) then
    if product(a[k][1], k=1..nops(a))=product(b[k][1], k=1..nops(a)) then print(n);
fi; fi; od; end:
A201009(100000); # Paolo P. Lava, Jan 09 2013

Prog

(Haskell)
a201009 = a201009_list
a201009_list = 1 : filter
   (\x -> a027748_row x == a027748_row (a003415 x)) [2..]
-- Reinhard Zumkeller, Jan 16 2013
(Python)
from sympy import primefactors, factorint
A201009 = [n for n in range(1, 10**5) if primefactors(n) == primefactors(sum([int(n*e/p) for p, e in factorint(n).items()]) if n > 1 else 0)] # Chai Wah Wu, Aug 21 2014

Crossrefs

Cf. A001221, A003415, A027748, A051674, A055744, A081377, A110751, A110819.

Sequence in context: A072653 A368107 A008478 * A111260 A067688 A097374

Adjacent sequences: A201006 A201007 A201008 * A201010 A201011 A201012

Keyword

nonn

Author

Paolo P. Lava, Jan 09 2013

Status

approved