A147581 Numbers with exactly 8 distinct odd prime divisors {3,5,7,11,13,17,19,23}.

111546435, 334639305, 557732175, 780825045, 1003917915, 1227010785, 1450103655, 1673196525, 1896289395, 2119382265, 2342475135, 2565568005, 2788660875, 3011753745, 3681032355, 3904125225, 4350310965, 5019589575, 5465775315, 5688868185, 6135053925, 6358146795

Offset: 1

Author

Artur Jasinski, Nov 07 2008

Keywords

• nonn

Comments

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

( Family of sequences for successive n odd primes )

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

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

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

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

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

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

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

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

Python

from sympy import integer_log
def A147581(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i23 in range(integer_log(x,23)[0]+1):
            for i19 in range(integer_log(x23:=x//23**i23,19)[0]+1):
                for i17 in range(integer_log(x19:=x23//19**i19,17)[0]+1):
                    for i13 in range(integer_log(x17:=x19//17**i17,13)[0]+1):
                        for i11 in range(integer_log(x13:=x17//13**i13,11)[0]+1):
                            for i7 in range(integer_log(x11:=x13//11**i11,7)[0]+1):
                                for i5 in range(integer_log(x7:=x11//7**i7,5)[0]+1):
                                    c -= integer_log(x7//5**i5,3)[0]+1
        return c
    return 111546435*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

Formula

Sum_{n>=1} 1/a(n) = 1/36495360. - Amiram Eldar, Dec 22 2020

Extensions

More terms from Amiram Eldar, Mar 11 2020