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%I A273773 #11 Jun 01 2016 21:40:23
%S A273773 43,67,101,137,163,181,229,241,281,313,353,421,433,487,563,601,617,
%T A273773 641,653,673,769,821,823,853,883,907,937,941,1009,1061,1093,1277,1303,
%U A273773 1361,1423,1429,1433,1447,1489,1549,1571,1579,1601,1607,1609,1613,1657,1667,1697,1741,1747
%N A273773 Primes p such that no practical number (A005153) exists between p and its successor.
%C A273773 According to Margenstern and proved by Weingartner (see links) the density of practical numbers is greater than the density of primes. Margenstern calculated that the density of practical numbers was approx 1.2767 (1.3411/1.059) times greater than the density of primes in the interval 1 to 10^12. This sequence shows that the set of places where no practical number exists between successive primes has a degree of regularity and appears to be infinite.
%H A273773 Frank M Jackson, <a href="/A273773/b273773.txt">Table of n, a(n) for n = 1..100000</a>
%H A273773 Maurice Margenstern, <a href="http://dx.doi.org/10.1016/S0022-314X(05)80022-8">Les nombres pratiques: théorie, observations et conjectures</a>, Journal of Number Theory 37 (1): 1-36, 1991.
%H A273773 A. Weingartner, <a href="http://qjmath.oxfordjournals.org/content/66/2/743">Practical numbers and the distribution of divisors</a>, The Quarterly Journal of Mathematics 66 (2): 743-758, 2015.
%H A273773 Wikipedia, <a href="https://en.wikipedia.org/wiki/Practical_number">Practical number</a>
%e A273773 a(6) = 181, the next prime is 191. In the integer interval [181, 191] there are no practical numbers. It is the 6th such occurrence.
%t A273773 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]]]; count[n_Integer] := Module[{t=0, m}, Do[If[PracticalQ[m], t++], {m, Prime[n], Prime[n + 1] - 1}]; t]; lst = {}; Do[If[count[n]==0, AppendTo[lst, Prime[n]]], {n, 1, 1000}];lst
%Y A273773 Cf. A005153.
%K A273773 nonn
%O A273773 1,1
%A A273773 _Frank M Jackson_, May 29 2016

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