%I #16 Oct 23 2024 00:43:02

%S 111546435,334639305,557732175,780825045,1003917915,1227010785,

%T 1450103655,1673196525,1896289395,2119382265,2342475135,2565568005,

%U 2788660875,3011753745,3681032355,3904125225,4350310965,5019589575,5465775315,5688868185,6135053925,6358146795

%N Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.

%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="/A147581/b147581.txt">Table of n, a(n) for n = 1..10000</a>

%F Sum_{n>=1} 1/a(n) = 1/36495360. - _Amiram Eldar_, Dec 22 2020

%t a = {}; Do[If[EulerPhi[111546435 x] == 36495360 x, AppendTo[a, 111546435 x]], {x, 1, 100}]; a

%o (Python)

%o from sympy import integer_log

%o def A147581(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 i23 in range(integer_log(x,23)[0]+1):

%o for i19 in range(integer_log(x23:=x//23**i23,19)[0]+1):

%o for i17 in range(integer_log(x19:=x23//19**i19,17)[0]+1):

%o for i13 in range(integer_log(x17:=x19//17**i17,13)[0]+1):

%o for i11 in range(integer_log(x13:=x17//13**i13,11)[0]+1):

%o for i7 in range(integer_log(x11:=x13//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 111546435*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

%E More terms from _Amiram Eldar_, Mar 11 2020