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.
1, 4, 27, 48, 72, 108, 162, 320, 800, 1792, 2000, 3125, 3840, 5000, 5760, 6272, 8640, 9600, 10935, 12500, 12960, 14400, 18225, 19440, 21504, 21600, 21952, 24000, 29160, 30375, 31250, 32256, 32400, 36000, 43740, 45056, 48384, 48600, 50625, 54000, 60000, 65610
Offset: 1
Keywords
Examples
320 is included because 320 = 2^6 * 5^1 and 2+5 = 6+1.
Links
- Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)
Programs
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Mathematica
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 *) 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 *) -
PARI
for(n=1,10^6,if(bigomega(n)==sumdiv(n,d,isprime(d)*d),print1(n,",")))
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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
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Sage
def d(n): 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)) return s0-s1 [k for k in (1..100000) if d(k)==0] # Giuseppe Coppoletta, May 07 2016
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