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%I #36 Dec 28 2018 23:25:35
%S 2,3,4,5,6,8,10,12,14,16,18,20,24,36,48,60,72,84,96,120
%N n such that all n - p are practical numbers where p is a practical number in range n/2 <= p < n.
%C The following is a quotation from Hage-Hassan in his paper (see Link below). "The (concept of) right and left symmetry is fundamental in physics. This incites us to ask whether this symmetry is in (the) primes. Find the numbers n with a + a' = n. a, a' are primes and {a} are all the primes with: n/2 <= a < n and n = 2,3, ..."
%C This sequence is analogous to A320447. Instead of the sequence of primes it uses the sequence of practical numbers (A005153). It is conjectured that the sequence is finite and full.
%H Mehdi Hage-Hassan, <a href="https://hal.archives-ouvertes.fr/hal-00879586/document">An elementary introduction to Quantum mechanic</a>, hal-00879586 2013 pp 58.
%e a(13)=24, because the practical numbers p in the range 12 <= p < 24 are {12, 16, 18, 20}. Also the complementary set {12, 8, 6, 4} has all its members practical numbers. This is the 13th occurrence of such a number.
%t 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 plst[n_] := Select[Range[Ceiling[n/2], n-1], PracticalQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], PracticalQ], AppendTo[lst, n]], {n, 1, 10000}]; lst
%Y Cf. A005153, A320447.
%K nonn,more
%O 1,1
%A _Frank M Jackson_, Dec 18 2018
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