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A181629 Positive integers k = p_1^{r_1} ... p_n^{r_n} such that sum_{i=1..n} p_i^{-r_i} >= 1 (Non-Hyperbolic Integers).

%I A181629 #14 Sep 08 2022 08:45:54
%S A181629 1,30,210,330,390,462,510,546,570,690,714,798,858,870,930,966,1050,
%T A181629 1110,1218,1230,1290,1302,1410,1470,1554,1590,1722,1770,1830,2010,
%U A181629 2130,2190,2310,2370,2490,2670,2730,2910,3030,3090,3210,3270,3390,3570,3630,3810
%N A181629 Positive integers k = p_1^{r_1} ... p_n^{r_n} such that sum_{i=1..n} p_i^{-r_i} >= 1 (Non-Hyperbolic Integers).
%C A181629 First odd term greater than 1 is 3234846615. - _Robert G. Wilson v_, Nov 04 2010
%C A181629 Also numbers n such that A028236(n)/n >= 1. - _Klaus Brockhaus_, Nov 06 2010
%e A181629 a(2) = 30, since 30 = 2*3*5 and 1/2 + 1/3 + 1/5 = 31/30 >= 1.
%t A181629 DeleteCases[ Table[k; A = FactorInteger[k]; If[Sum[1/A[[j]][[1]]^A[[j]][[2]], {j, 1, Length[A]}] >= 1, k, 0], {k, 1, 3900}], 0]
%t A181629 fQ[n_] := Block[{fi = Transpose@ FactorInteger@ n}, Plus @@ (1/(First@fi ^ Last@fi)) >= 1]; Select[Range@ 3900, fQ] (* _Robert G. Wilson v_, Nov 04 2010 *)
%o A181629 (Magma) [1] cat [ k: k in [2..4000] | &+[ f[i, 1]^-f[i, 2]: i in [1..#f] ] ge 1 where f is Factorization(k) ]; // _Klaus Brockhaus_, Nov 06 2010
%Y A181629 Cf. A028236 (if n = Product (p_j^k_j), a(n) = numerator of Sum 1/p_j^k_j). - _Klaus Brockhaus_, Nov 06 2010
%K A181629 nonn
%O A181629 1,2
%A A181629 _Roberto E. Martinez II_, Nov 02 2010, Nov 05 2010