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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
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
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.)