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.

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

Offset

1,1

Comments

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, ..."

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.

Links

Table of n, a(n) for n=1..12.

Mehdi Hage-Hassan, An elementary introduction to Quantum mechanic, hal-00879586 2013 pp 58.

Example

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.

Mathematica

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]]];
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];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
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

Crossrefs

Cf. A267124, A320447, A321152.

Sequence in context: A340111 A211659 A301806 * A066847 A370121 A057887

Adjacent sequences: A374054 A374055 A374056 * A374058 A374059 A374060

Keyword

nonn,more

Author

Frank M Jackson, Jun 26 2024

Status

approved