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A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).

%I A272858 #45 Mar 24 2020 03:30:35
%S A272858 1,4,27,72,96,108,486,800,1280,3125,6272,10976,12500,14336,21600,
%T A272858 28800,30375,34560,36000,38880,48600,54675,84375,92160,96000,121500,
%U A272858 134456,153600,169344,217728,218700,225000,247808,262440,296352,300000,337500,340736,387072,395136,489888,666792,703125,750141,781250,823543,857304,885735
%N A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).
%C A272858 A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Similarly, 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.
%C A272858 The condition defining this sequence coincides with the condition in A272859 at least for the terms of A114129.
%H A272858 Giuseppe Coppoletta and Giovanni Resta, <a href="/A272858/b272858.txt">Table of n, a(n) for n = 1..6058</a> (terms < 10^18, first 100 terms from G. Coppoletta)
%F A272858 If N is a positive integer and N = Product_{i=1..k} (p_i)^e_i is its prime factorization, then N is in A272858 iff Product_{i=1..k} (1 + p_i) = Product_{i=1..k} (1 + e_i).
%F A272858 For a number with three different prime factors N = p1^e1 * p2^e2 * p3^e3, the defining condition can be expressed as: p1 + p2 + p3 + p1*p2 + p1*p3 + p2*p3 + p1*p2*p3 = e1 + e2 + e3 + e1*e2 + e1*e3 + e2*e3 + e1*e2*e3.
%e A272858 92160 is included because 92160 = 2^11 * 3^2 * 5 and (2+1)*(3+1)*(5+1) = (11+1)*(2+1)*(1+1).
%t A272858 ok[n_] := Block[{p,e}, {p,e} = Transpose@ FactorInteger@ n; Times @@ (1+p) == Times @@ (1+e)]; Select[Range[10^6], ok] (* _Giovanni Resta_, May 08 2016 *)
%o A272858 (Sage)
%o A272858 def d(n):
%o A272858     v = factor(n)
%o A272858     d1 = prod(1 + w[0] for w in v)
%o A272858     d2 = prod(1 + w[1] for w in v)
%o A272858     return d1 == d2
%o A272858 [k for k in (1..10000) if d(k)]
%o A272858 (PARI) is(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]+1)==prod(i=1,#f~,f[i,2]) \\ _Charles R Greathouse IV_, Sep 08 2016
%Y A272858 Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272859.
%K A272858 nonn
%O A272858 1,2
%A A272858 _Giuseppe Coppoletta_, May 08 2016