A374057 - OEIS

%I #7 Jul 07 2024 21:09:32

%S 2,3,4,7,8,12,21,22,26,62,72,182

%N 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 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 primitive practical numbers (A267124). 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 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 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 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 PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);

%t 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 Cf. A267124, A320447, A321152.

%K nonn,more

%O 1,1

%A _Frank M Jackson_, Jun 26 2024