cp's OEIS Frontend

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'.

%I A201009 #28 May 09 2021 13:34:09
%S A201009 1,4,16,27,108,144,256,432,500,784,972,1323,1728,2700,2916,3125,3456,
%T A201009 5292,8788,11664,12500,13068,15376,16875,19683,20736,23328,25000,
%U A201009 27648,28125,31212,34300,47916,54000,57132,65536,72000,78732,97556,102400,103788,104544
%N 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'.
%C A201009 A027748(n,k) = A027748(A003415(n),k) for k=1..A001221(n). - _Reinhard Zumkeller_, Jan 16 2013
%C A201009 A051674 is a subsequence of this sequence.
%H A201009 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 A201009 n = 1728 = 2^6*3^3, n' = 6912 = 2^8*3^3 have the same prime factors 2 and 3.
%p A201009 with(numtheory);
%p A201009 A201009:=proc(q)
%p A201009 local a,b,k,n;
%p A201009 for n from 1 to q do
%p A201009   a:=ifactors(n)[2]; b:=ifactors(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]))[2];
%p A201009   if nops(a)=nops(b) then
%p A201009     if product(a[k][1],k=1..nops(a))=product(b[k][1],k=1..nops(a)) then print(n);
%p A201009 fi; fi; od; end:
%p A201009 A201009(100000); # _Paolo P. Lava_, Jan 09 2013
%o A201009 (Haskell)
%o A201009 a201009 = a201009_list
%o A201009 a201009_list = 1 : filter
%o A201009    (\x -> a027748_row x == a027748_row (a003415 x)) [2..]
%o A201009 -- _Reinhard Zumkeller_, Jan 16 2013
%o A201009 (Python)
%o A201009 from sympy import primefactors, factorint
%o A201009 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 A201009 Cf. A001221, A003415, A027748, A051674, A055744, A081377, A110751, A110819.
%K A201009 nonn
%O A201009 1,2
%A A201009 _Paolo P. Lava_, Jan 09 2013