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A321152 n such that all n - p are practical numbers where p is a practical number in range n/2 <= p < n. 0
2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20, 24, 36, 48, 60, 72, 84, 96, 120 (list; graph; refs; listen; history; text; internal format)
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 practical numbers (A005153). It is conjectured that the sequence is finite and full.
LINKS
Mehdi Hage-Hassan, An elementary introduction to Quantum mechanic, hal-00879586 2013 pp 58.
EXAMPLE
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.
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]]];
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
CROSSREFS
Sequence in context: A005377 A120370 A011866 * A292983 A174709 A008724
KEYWORD
nonn,more
AUTHOR
Frank M Jackson, Dec 18 2018
STATUS
approved

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)