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Numbers with exactly 4 distinct prime divisors {2,3,5,7}.
16

%I #38 Nov 05 2024 14:48:12

%S 210,420,630,840,1050,1260,1470,1680,1890,2100,2520,2940,3150,3360,

%T 3780,4200,4410,5040,5250,5670,5880,6300,6720,7350,7560,8400,8820,

%U 9450,10080,10290,10500,11340,11760,12600,13230,13440,14700,15120,15750,16800

%N Numbers with exactly 4 distinct prime divisors {2,3,5,7}.

%C Successive numbers k such that EulerPhi(x)/x = m:

%C ( Family of sequences for successive n primes )

%C m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079

%C m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845

%C m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207

%C m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571

%C m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572

%C m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573

%C m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574

%C m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575

%H David A. Corneth, <a href="/A147571/b147571.txt">Table of n, a(n) for n = 1..10688</a>

%F a(n) = 210 * A002473(n). - _David A. Corneth_, May 14 2019

%F Sum_{n>=1} 1/a(n) = 1/48. - _Amiram Eldar_, Nov 12 2020

%t a = {}; Do[If[EulerPhi[x]/x == 8/35, AppendTo[a, x]], {x, 1, 100000}]; a

%t Select[Range[20000],PrimeNu[#]==4&&Max[FactorInteger[#][[;;,1]]]<11&] (* _Harvey P. Dale_, Nov 05 2024 *)

%o (Magma) [n: n in [1..2*10^4] | PrimeDivisors(n) eq [2,3,5,7]]; // _Vincenzo Librandi_, Sep 15 2015

%o (PARI) is(n)=n%210==0 && n==2^valuation(n,2) * 3^valuation(n,3) * 5^valuation(n,5) * 7^valuation(n,7) \\ _Charles R Greathouse IV_, Jun 19 2016

%Y Cf. A002473, A086780, A143207, A147572, A147573, A147574, A147575, A147576, A147577, A147578, A147579, A147580.

%K nonn

%O 1,1

%A _Artur Jasinski_, Nov 07 2008