A231231 Numbers n such that, in the prime factorization of n, the product of the exponents equals the sum of prime factors and exponents.

432, 648, 1152, 4000, 5400, 8748, 9000, 12800, 12960, 13500, 17280, 19440, 21952, 25000, 48000, 48384, 50625, 60000, 78400, 87480, 100352, 114048, 150000, 189000, 202176, 263424, 303264, 303750, 304128, 340736, 356400, 367416, 368640, 370440, 374544, 384912

Offset

1,1

Comments

If n = p_1^c_1 * p_2^c_2 * p_3^c_3 * ... * p_k^c_k, where c's are positive integers and p's are distinct primes, then product{j=1 to k}[c_j] = sum{j=1 to k}[p_j+c_j].

Links

Table of n, a(n) for n=1..36.

Example

9000 = 3^2 * 2^3 * 5^3. Product of exponents is 2*3*3=18, sum of prime factors and exponents is 3+2+2+3+5+3=18, hence 9000 is in the sequence.

Mathematica

t = {}; n = 1; While[Length[t] < 38, n++; f = FactorInteger[n]; sm = Total[Flatten[f]]; pr = Times @@ Transpose[f][[2]]; If[sm == pr, AppendTo[t, n]]]; t (* T. D. Noe, Nov 08 2013 *)
peQ[n_]:=Module[{fi=FactorInteger[n]}, Times@@fi[[All, 2]]==Total[ Flatten[ fi]]]; Select[Range[400000], peQ] (* Harvey P. Dale, May 21 2019 *)

Crossrefs

Cf. A054411, A054412, A071174, A071175, A122406, A231293.

Sequence in context: A245469 A069781 A337670 * A179666 A203663 A250815

Adjacent sequences: A231228 A231229 A231230 * A231232 A231233 A231234

Keyword

nonn

Author

Alex Ratushnyak, Nov 06 2013

Status

approved