A175755 Numbers with 49 divisors.

46656, 1000000, 7529536, 11390625, 85766121, 113379904, 308915776, 1291467969, 1544804416, 1838265625, 3010936384, 3518743761, 9474296896, 17596287801, 27680640625, 34296447249, 38068692544, 56800235584, 75418890625, 107918163081, 164206490176, 208422380089

Offset

1,1

Comments

Numbers of the forms p^48 and p^6*q^6, where p and q are distinct primes.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

OEIS Wiki, Index entries for number of divisors

Formula

A000005(a(n)) = 49.

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

Example

a(1) = A114334(49); a(2) = A159765(49).

Mathematica

Select[Range[100000000], DivisorSigma[0, #]==48&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

Prog

(Haskell)
a175755 n = a175755_list !! (n-1)
a175755_list = m (map (^ 48) a000040_list) (map (^ 6) a006881_list) where
   m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
                           | otherwise = y : m xs' ys
-- Reinhard Zumkeller, Nov 29 2011
(PARI) is(n)=numdiv(n)==49 \\ Charles R Greathouse IV, Jun 19 2016
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A175755(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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]))
    return bisection(f, n, n) # Chai Wah Wu, Feb 22 2025

Crossrefs

Cf. A000005, A139575, A175754, A201266, A137488, A135581.

Sequence in context: A057068 A223510 A251190 * A223562 A016866 A016914

Adjacent sequences: A175752 A175753 A175754 * A175756 A175757 A175758

Keyword

nonn

Author

Jaroslav Krizek, Aug 27 2010

Extensions

Extended by T. D. Noe, May 08 2011

Status

approved