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

1,2

Comments

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

Links

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)

Example

320 is included because 320 = 2^6 * 5^1 and 2+5 = 6+1.

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 *)

Prog

(PARI) for(n=1, 10^6, if(bigomega(n)==sumdiv(n, d, isprime(d)*d), print1(n, ", ")))
(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
(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

Crossrefs

Cf. A054412, A068935, A068936, A068937, A068938.

Sequence in context: A239283 A082872 A274854 * A051506 A370492 A347671

Adjacent sequences: A054408 A054409 A054410 * A054412 A054413 A054414

Keyword

nonn

Author

Leroy Quet, May 09 2000

Status

approved