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A231231 Numbers n such that, in the prime factorization of n, the product of the exponents equals the sum of prime factors and exponents. 1

%I #16 May 21 2019 14:54:42

%S 432,648,1152,4000,5400,8748,9000,12800,12960,13500,17280,19440,21952,

%T 25000,48000,48384,50625,60000,78400,87480,100352,114048,150000,

%U 189000,202176,263424,303264,303750,304128,340736,356400,367416,368640,370440,374544,384912

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

%C 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].

%e 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.

%t 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 *)

%t peQ[n_]:=Module[{fi=FactorInteger[n]},Times@@fi[[All,2]]==Total[ Flatten[ fi]]]; Select[Range[400000],peQ] (* _Harvey P. Dale_, May 21 2019 *)

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

%K nonn

%O 1,1

%A _Alex Ratushnyak_, Nov 06 2013

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Last modified March 28 16:12 EDT 2024. Contains 371254 sequences. (Running on oeis4.)