A147572 - OEIS

%I #21 Sep 18 2024 05:53:48

%S 2310,4620,6930,9240,11550,13860,16170,18480,20790,23100,25410,27720,

%T 32340,34650,36960,41580,46200,48510,50820,55440,57750,62370,64680,

%U 69300,73920,76230,80850,83160,92400,97020,101640,103950,110880,113190,115500,124740,127050

%N Numbers with exactly 5 distinct prime divisors {2,3,5,7,11}.

%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 Amiram Eldar, <a href="/A147572/b147572.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 2310 * A051038(n). - _Amiram Eldar_, Mar 10 2020

%F Sum_{n>=1} 1/a(n) = 1/480. - _Amiram Eldar_, Nov 12 2020

%t a = {}; Do[If[EulerPhi[x]/x == 16/77, AppendTo[a, x]], {x, 1, 100000}]; a

%t Select[Range[130000],FactorInteger[#][[All,1]]=={2,3,5,7,11}&] (* _Harvey P. Dale_, Oct 04 2020 *)

%o (Python)

%o from sympy import integer_log, prevprime

%o def A147572(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))

%o     def f(x): return n+x-g(x,11)

%o     return 2310*bisection(f,n,n) # _Chai Wah Wu_, Sep 16 2024

%Y Cf. A051038, A060735, A143207, A147571-A147575, A147576-A147580.

%K nonn

%O 1,1

%A _Artur Jasinski_, Nov 07 2008

%E More terms from _Amiram Eldar_, Mar 10 2020