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A287681 Twin practical numbers: numbers k such that both k and k+2 are practical numbers.

%I A287681 #15 Apr 04 2024 04:45:00
%S A287681 2,4,6,16,18,28,30,40,54,64,78,88,126,160,196,198,208,270,304,306,340,
%T A287681 378,390,414,448,460,462,510,520,544,558,700,702,726,798,810,858,868,
%U A287681 880,918,928,966,990,1024,1120,1216,1230,1240,1288,1300,1350,1408,1456
%N A287681 Twin practical numbers: numbers k such that both k and k+2 are practical numbers.
%C A287681 Melfi proved that this sequence is infinite.
%H A287681 Amiram Eldar, <a href="/A287681/b287681.txt">Table of n, a(n) for n = 1..10000</a>
%H A287681 Giuseppe Melfi, <a href="http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/201">A note on twin practical numbers</a>, Le Matematiche, Vol. 57 No. 1 (2004), pp. 111-117.
%H A287681 Giuseppe Melfi, <a href="http://members.unine.ch/giuseppe.melfi/articoli/smapoto.pdf">A survey on practical numbers</a>, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.
%H A287681 Giuseppe Melfi, <a href="http://members.unine.ch/giuseppe.melfi/twin.html">Twin practical numbers</a>.
%t A287681 practicalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
%t A287681 a={}; p1=False; k=2; While[Length[a]<100, p2=practicalQ[k]; If[p1 && p2, a=AppendTo[a,k-2]]; p1 = p2; k+=2];a
%Y A287681 Cf. A005153, A210479, A208249, A209236, A210480, A209236, A287682, A287683.
%K A287681 nonn
%O A287681 1,1
%A A287681 _Amiram Eldar_, May 29 2017