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 A273773 Primes p such that no practical number (A005153) exists between p and its successor. 3
 43, 67, 101, 137, 163, 181, 229, 241, 281, 313, 353, 421, 433, 487, 563, 601, 617, 641, 653, 673, 769, 821, 823, 853, 883, 907, 937, 941, 1009, 1061, 1093, 1277, 1303, 1361, 1423, 1429, 1433, 1447, 1489, 1549, 1571, 1579, 1601, 1607, 1609, 1613, 1657, 1667, 1697, 1741, 1747 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Frank M Jackson, Table of n, a(n) for n = 1..100000 Maurice Margenstern, Les nombres pratiques: théorie, observations et conjectures, Journal of Number Theory 37 (1): 1-36, 1991. A. Weingartner, Practical numbers and the distribution of divisors, The Quarterly Journal of Mathematics 66 (2): 743-758, 2015. Wikipedia, Practical number EXAMPLE 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. 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]]]; 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 CROSSREFS Cf. A005153. Sequence in context: A102269 A020349 A050959 * A139917 A146334 A039385 Adjacent sequences: A273770 A273771 A273772 * A273774 A273775 A273776 KEYWORD nonn AUTHOR Frank M Jackson, May 29 2016 STATUS approved

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Last modified July 18 09:18 EDT 2024. Contains 374378 sequences. (Running on oeis4.)