A231293 Numbers n such that, in the prime factorization of n, the product of the prime factors equals the sum of prime factors and exponents.

20, 50, 112, 392, 1372, 2816, 3645, 4802, 6075, 10125, 13312, 15488, 16875, 28125, 46875, 85184, 86528, 278528, 413343, 468512, 562432, 964467, 1245184, 2250423, 2367488, 2576816, 3655808, 3932160, 5250987, 5898240, 8847360, 9830400, 11829248, 12252303

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}[p_j] = sum{j=1 to k}[p_j+c_j].

Links

Table of n, a(n) for n=1..34.

Example

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.
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.
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.

Mathematica

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
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 *)

Crossrefs

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

Sequence in context: A304832 A007589 A182462 * A227771 A228023 A241609

Adjacent sequences: A231290 A231291 A231292 * A231294 A231295 A231296

Keyword

nonn

Author

Alex Ratushnyak, Nov 06 2013

Status

approved