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%I #14 Jun 15 2022 12:46:46
%S 8,16,24,27,32,36,48,54,64,72,80,81,96,100,108,112,125,128,135,144,
%T 160,162,180,189,192,196,200,216,224,225,240,243,250,256,270,288,300,
%U 320,324,336,343,352,360,375,378,384,392,400,405,416,432,441,448,450,480
%N Numbers n such that n/rad(n) is greater than the greatest prime dividing n.
%C n such that n/rad(n)> gpf(n); numbers n such that n/A007947(n) > A006530(n) where A007947 is the product of the distinct prime factors of n and A006530 is the greatest prime dividing n.
%C The sequence A137845 (logarithmically smooth numbers)is included in this sequence.
%C It appears that there exists consecutive numbers such that (80,81), (224,225), (675,676), (1088,1089), (1215,1216), (2375,2376), (2400,2401), (2600, 2601), (3024,3025), (3249,3250), (3968,3969), (4224,4225), (4374,4375), (5831,5832),...
%C But it appears also that (2400,2401) and (4374,4375)are the only consecutive numbers in the sequence A137845.
%H Michel Lagneau, <a href="/A228957/b228957.txt">Table of n, a(n) for n = 1..10000</a>
%e 24 is in the sequence because the prime divisors of 24 are 2 and 3 and 24/2*3 > 3.
%p with(numtheory) :for n from 1 to 400 do:x:=factorset(n):n1:=nops(x): p:= product('x[i]', 'i'=1..n1):m:=n/p:if m> x[n1]then printf(`%d, `,n):else fi:od:
%t rad[n_]:=Times@@(First@#&/@FactorInteger@n);Select[Range[2,1000],FactorInteger[#][[-1,1]]<#/rad[#]&]
%t nrQ[n_]:=Module[{x=FactorInteger[n][[All,1]]},n/Times@@x>Last[x]]; Select[Range[ 500],nrQ] (* _Harvey P. Dale_, Jun 15 2022 *)
%o (PARI) is(n)=my(f=factor(n)); prod(i=1,#f~,f[i,1]^(f[i,2]-1)) > f[#f~,1] \\ _Charles R Greathouse IV_, Sep 09 2013
%Y Cf. A006530, A007947, A137845.
%K nonn
%O 1,1
%A _Michel Lagneau_, Sep 09 2013
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