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A374057 Integers k such that all k - p are primitive practical numbers where p is a primitive practical number in range k/2 <= p < k.

%I A374057 #7 Jul 07 2024 21:09:32
%S A374057 2,3,4,7,8,12,21,22,26,62,72,182
%N A374057 Integers k such that all k - p are primitive practical numbers where p is a primitive practical number in range k/2 <= p < k.
%C A374057 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 A374057 This sequence is analogous to A320447. Instead of the sequence of primes it uses the sequence of primitive practical numbers (A267124). It is conjectured that the sequence is finite and full.
%H A374057 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 A374057 182 is a term because the primitive practical numbers p in the range 91 <= p < 182 are {104, 140}. Also the complementary set {78, 42} has all its members primitive practical numbers.
%t A374057 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 A374057 DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[! PracticalQ[n/plst[[m]]], ok=True, ok=False; Break[]], {m, 1, Length@plst}]; ok];
%t A374057 PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
%t A374057 plst[n_] := Select[Range[Ceiling[n/2], n-1], PPracticalQ]; lst={}; Do[If[plst[n]!={}&&AllTrue[n-plst[n], PPracticalQ], AppendTo[lst, n]], {n, 1, 10000}]; lst
%Y A374057 Cf. A267124, A320447, A321152.
%K A374057 nonn,more
%O A374057 1,1
%A A374057 _Frank M Jackson_, Jun 26 2024