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A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.

%I A054411 #38 Jan 09 2021 10:40:23
%S A054411 1,4,27,48,72,108,162,320,800,1792,2000,3125,3840,5000,5760,6272,8640,
%T A054411 9600,10935,12500,12960,14400,18225,19440,21504,21600,21952,24000,
%U A054411 29160,30375,31250,32256,32400,36000,43740,45056,48384,48600,50625,54000,60000,65610
%N A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.
%C A054411 Numbers for which the sum of distinct prime factors equals the sum of exponents in the prime factorization, A008472(n)=A001222(n). - _Reinhard Zumkeller_, Mar 08 2002
%H A054411 Giuseppe Coppoletta and Giovanni Resta, <a href="/A054411/b054411.txt">Table of n, a(n) for n = 1..10000</a> (first 100 terms from G. Coppoletta)
%e A054411 320 is included because 320 = 2^6 * 5^1 and 2+5 = 6+1.
%t A054411 f[n_]:=Plus@@First/@FactorInteger[n]==Plus@@Last/@FactorInteger[n]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,0,3*8!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 19 2010 *)
%t A054411 max = 10^12; Sort@Reap[Sow@1; Do[p = Select[IntegerPartitions[se, All, Prime@ Range@ PrimePi@ se], Sort[#] == Union[#] &]; Do[ np = Length[f]; va = IntegerPartitions[se, {np}, Range[se]]; Do[pe = Permutations[v]; Do[z = Times @@ (f^e); If[z <= max, Sow@z], {e, pe}], {v, va}], {f, p}], {se, 2, Log2[max]}]][[2, 1]] (* _Giovanni Resta_, May 07 2016 *)
%o A054411 (PARI) for(n=1,10^6,if(bigomega(n)==sumdiv(n,d,isprime(d)*d),print1(n,",")))
%o A054411 (PARI) is(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]-f[i,2])==0 \\ _Charles R Greathouse IV_, Sep 08 2016
%o A054411 (Sage) def d(n):
%o A054411     v=factor(n)[:]; L=len(v); s0=sum(v[j][0] for j in range(L)); s1=sum(v[j][1] for j in range(L))
%o A054411     return s0-s1
%o A054411 [k for k in (1..100000) if d(k)==0] # _Giuseppe Coppoletta_, May 07 2016
%Y A054411 Cf. A054412, A068935, A068936, A068937, A068938.
%K A054411 nonn
%O A054411 1,2
%A A054411 _Leroy Quet_, May 09 2000