A272818 - OEIS

%I #44 Mar 24 2020 03:25:24

%S 1,4,27,72,96,108,486,800,1280,3125,6272,10976,12500,14336,21600,

%T 30375,36000,48600,51840,54675,69120,84375,121500,134456,169344,

%U 174960,192000,225000,240000,247808,337500,340736,395136,435456,451584,703125,750141,781250,787320,823543,857304,885735

%N Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization.

%C For p prime, p^p satisfy the condition, hence A051674 (and also A048102) is a subsequence. Moreover, if p and q are primes and i and j are positive integers, if p^i * q^j verify the condition, then the same is true for p^j * q^i. So A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e. any permutation of its prime signature) gives also a term. In addition, any number having no more than two distinct prime factors (apart their multiplicity) is a term iff it belongs also to A272858.

%H Giuseppe Coppoletta and Giovanni Resta, <a href="/A272818/b272818.txt">Table of n, a(n) for n = 1..1413</a> (terms < 10^19, first 100 terms from G. Coppoletta)

%e 885735 = 3^11 * 5 is included because (3+5) + 3*5 = (11+1) + 11*1.

%e 2^10 * 3^6 * 19^2 is included because (2+3+19)+ 2*3*19 = (10+6+2)+ 10*6*2.

%t Select[Range[10^6], Total@ First@ # + Times @@ First@ # == Total@ Last@ # + Times @@ Last@ # &@ Transpose@ FactorInteger@ # &] (* _Michael De Vlieger_, May 08 2016 *)

%o (Sage)

%o def d(n):

%o     v = factor(n)

%o     d1 = sum(w[0] for w in v) + prod(w[0] for w in v)

%o     d2 = sum(w[1] for w in v) + prod(w[1] for w in v)

%o     return d1 == d2

%o [k for k in (1..10000) if d(k)]

%o (PARI) spp(v) = vecsum(v) + prod(k=1, #v, v[k]);

%o isok(n) = my(f = factor(n)); spp(f[,1]) == spp(f[,2]); \\ _Michel Marcus_, May 08 2016

%Y Cf. A048102, A054411, A054412, A071174, A122406, A272858, A272859.

%K nonn

%O 1,2

%A _Giuseppe Coppoletta_, May 08 2016