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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'.
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%I #28 May 09 2021 13:34:09

%S 1,4,16,27,108,144,256,432,500,784,972,1323,1728,2700,2916,3125,3456,

%T 5292,8788,11664,12500,13068,15376,16875,19683,20736,23328,25000,

%U 27648,28125,31212,34300,47916,54000,57132,65536,72000,78732,97556,102400,103788,104544

%N 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'.

%C A027748(n,k) = A027748(A003415(n),k) for k=1..A001221(n). - _Reinhard Zumkeller_, Jan 16 2013

%C A051674 is a subsequence of this sequence.

%H Paolo P. Lava and Donovan Johnson, <a href="/A201009/b201009.txt">Table of n, a(n) for n = 1..500</a> (first 100 terms from Paolo P. Lava)

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

%p with(numtheory);

%p A201009:=proc(q)

%p local a,b,k,n;

%p for n from 1 to q do

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

%p if nops(a)=nops(b) then

%p if product(a[k][1],k=1..nops(a))=product(b[k][1],k=1..nops(a)) then print(n);

%p fi; fi; od; end:

%p A201009(100000); # _Paolo P. Lava_, Jan 09 2013

%o (Haskell)

%o a201009 = a201009_list

%o a201009_list = 1 : filter

%o (\x -> a027748_row x == a027748_row (a003415 x)) [2..]

%o -- _Reinhard Zumkeller_, Jan 16 2013

%o (Python)

%o from sympy import primefactors, factorint

%o 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

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

%K nonn

%O 1,2

%A _Paolo P. Lava_, Jan 09 2013