

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
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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



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



KEYWORD

nonn


AUTHOR



STATUS

approved



