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Numbers k such that the product of the logarithms of k's prime factors is greater than their sum.
1

%I #28 Dec 14 2018 20:19:02

%S 77,85,91,95,115,119,121,133,143,145,155,161,169,185,187,203,205,209,

%T 215,217,221,235,245,247,253,259,265,275,287,289,295,299,301,305,319,

%U 323,325,329,335,341,343,355,357,361,363,365,371,377,385,391,395,399

%N Numbers k such that the product of the logarithms of k's prime factors is greater than their sum.

%C First even term is 1334.

%C Previous name was: If p^a(p,n) is highest power of p that divides n, then Product_(p=primes) [log(p)^a(p,n) ] > log(n).

%C From _Charlie Neder_, Dec 03 2018: (Start)

%C Primes are counted with multiplicity.

%C Does this sequence have asymptotic density 1?

%C a(100000) = 189835. (End)

%H Charlie Neder, <a href="/A052202/b052202.txt">Table of n, a(n) for n = 1..1000</a>

%e 245 is included because 245 = 5*7^2 and log(5)*log(7)^2 > log(245).

%t aQ[n_] := Times@@Power@@@({Log[#1], #2} & @@@FactorInteger[n]) > Log[n]; Select[Range[400], aQ] (* _Amiram Eldar_, Dec 03 2018 *)

%o (PARI) isok(n) = my(f=factor(n)); prod(k=1, #f~, log(f[k,1])^f[k,2]) > log(n); \\ _Michel Marcus_, Dec 04 2018

%K nonn

%O 1,1

%A _Leroy Quet_, Jan 28 2000

%E Title changed by _Charlie Neder_, Dec 04 2018