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Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.
7

%I #16 Oct 22 2024 20:17:07

%S 1155,3465,5775,8085,10395,12705,17325,24255,28875,31185,38115,40425,

%T 51975,56595,63525,72765,86625,88935,93555,114345,121275,139755,

%U 144375,155925,169785,190575,202125,218295,259875,266805,280665,282975,317625

%N Numbers with exactly 4 distinct odd prime divisors {3,5,7,11}.

%C Numbers k such that phi(k)/k = m

%C ( Family of sequences for successive n odd primes )

%C m=2/3 numbers with exactly 1 distinct prime divisor {3} see A000244

%C m=8/15 numbers with exactly 2 distinct prime divisors {3,5} see A033849

%C m=16/35 numbers with exactly 3 distinct prime divisors {3,5,7} see A147576

%C m=32/77 numbers with exactly 4 distinct prime divisors {3,5,7,11} see A147577

%C m=384/1001 numbers with exactly 5 distinct prime divisors {3,5,7,11,13} see A147578

%C m=6144/17017 numbers with exactly 6 distinct prime divisors {3,5,7,11,13,17} see A147579

%C m=3072/323323 numbers with exactly 7 distinct prime divisors {3,5,7,11,13,17,19} see A147580

%C m=110592/323323 numbers with exactly 8 distinct prime divisors {3,5,7,11,13,17,19,23} see A147581

%H Amiram Eldar, <a href="/A147577/b147577.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = 1/480. - _Amiram Eldar_, Dec 22 2020

%t a = {}; Do[If[EulerPhi[x]/x == 32/77, AppendTo[a, x]], {x, 1, 1000000}]; a

%t Select[Range[350000],EulerPhi[#]/#==32/77&] (* _Harvey P. Dale_, Mar 25 2016 *)

%o (Python)

%o from sympy import integer_log

%o def A147577(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 f(x):

%o c = n+x

%o for i11 in range(integer_log(x,11)[0]+1):

%o for i7 in range(integer_log(x11:=x//11**i11,7)[0]+1):

%o for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):

%o c -= integer_log(x7//5**i5,3)[0]+1

%o return c

%o return 1155*bisection(f,n,n) # _Chai Wah Wu_, Oct 22 2024

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

%K nonn

%O 1,1

%A _Artur Jasinski_, Nov 07 2008