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A175755 Numbers with 49 divisors.

%I A175755 #30 Feb 22 2025 12:18:55
%S A175755 46656,1000000,7529536,11390625,85766121,113379904,308915776,
%T A175755 1291467969,1544804416,1838265625,3010936384,3518743761,9474296896,
%U A175755 17596287801,27680640625,34296447249,38068692544,56800235584,75418890625,107918163081,164206490176,208422380089
%N A175755 Numbers with 49 divisors.
%C A175755 Numbers of the forms p^48 and p^6*q^6, where p and q are distinct primes.
%H A175755 T. D. Noe, <a href="/A175755/b175755.txt">Table of n, a(n) for n = 1..1000</a>
%H A175755 OEIS Wiki, <a href="https://oeis.org/wiki/Index_entries_for_number_of_divisors">Index entries for number of divisors</a>
%F A175755 A000005(a(n)) = 49.
%F A175755 Sum_{n>=1} 1/a(n) = (P(6)^2 - P(12))/2 + P(48) = 0.0000226806..., where P is the prime zeta function. - _Amiram Eldar_, Jul 03 2022
%e A175755 a(1) = A114334(49); a(2) = A159765(49).
%t A175755 Select[Range[100000000],DivisorSigma[0,#]==48&] (* _Vladimir Joseph Stephan Orlovsky_, May 06 2011 *)
%o A175755 (Haskell)
%o A175755 a175755 n = a175755_list !! (n-1)
%o A175755 a175755_list = m (map (^ 48) a000040_list) (map (^ 6) a006881_list) where
%o A175755    m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
%o A175755                            | otherwise = y : m xs' ys
%o A175755 -- _Reinhard Zumkeller_, Nov 29 2011
%o A175755 (PARI) is(n)=numdiv(n)==49 \\ _Charles R Greathouse IV_, Jun 19 2016
%o A175755 (Python)
%o A175755 from math import isqrt
%o A175755 from sympy import primepi, integer_nthroot, primerange
%o A175755 def A175755(n):
%o A175755     def bisection(f,kmin=0,kmax=1):
%o A175755         while f(kmax) > kmax: kmax <<= 1
%o A175755         kmin = kmax >> 1
%o A175755         while kmax-kmin > 1:
%o A175755             kmid = kmax+kmin>>1
%o A175755             if f(kmid) <= kmid:
%o A175755                 kmax = kmid
%o A175755             else:
%o A175755                 kmin = kmid
%o A175755         return kmax
%o A175755     def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,6)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(integer_nthroot(x,48)[0]))
%o A175755     return bisection(f,n,n) # _Chai Wah Wu_, Feb 22 2025
%Y A175755 Cf. A000005, A139575, A175754, A201266, A137488, A135581.
%K A175755 nonn
%O A175755 1,1
%A A175755 _Jaroslav Krizek_, Aug 27 2010
%E A175755 Extended by _T. D. Noe_, May 08 2011