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

%I #11 Aug 17 2016 18:25:28

%S 20,50,112,392,1372,2816,3645,4802,6075,10125,13312,15488,16875,28125,

%T 46875,85184,86528,278528,413343,468512,562432,964467,1245184,2250423,

%U 2367488,2576816,3655808,3932160,5250987,5898240,8847360,9830400,11829248,12252303

%N Numbers n such that, in the prime factorization of n, the product of the prime factors 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}[p_j] = sum{j=1 to k}[p_j+c_j].

%e 50 = 2 * 5^2; the product of the prime factors is 2 * 5 = 10, the sum of the prime factors and exponents is 2 + 1 + 5 + 2 = 10, hence 50 is in the sequence.

%e 112 = 2^4 * 7; the product of the prime factors is 2 * 7 = 14, the sum of the prime factors and exponents is 2 + 4 + 7 + 1 = 14, hence 112 is in the sequence.

%e 14172488 = 2^3 * 11^6, product of prime factors is 2*11 = 22, sum of prime factors and exponents is 2 + 3 + 11 + 6 = 22, hence 14172488 is in the sequence.

%t t = {}; n = 1; While[Length[t] < 30, n++; f = FactorInteger[n]; sm = Total[Flatten[f]]; pr = Times @@ Transpose[f][[1]]; If[sm == pr, AppendTo[t, n]]]; t

%t ppfQ[n_]:=Module[{f=FactorInteger[n]},Times@@[f][[All,1]] == Total[ Flatten[f]]]; Select[Range[13*10^6],ppfQ] (* _Harvey P. Dale_, Aug 17 2016 *)

%Y Cf. A054411, A054412, A071174, A071175, A122406, A231231.

%K nonn

%O 1,1

%A _Alex Ratushnyak_, Nov 06 2013