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

Author

Jaroslav Krizek, Aug 27 2010

Keywords

• nonn

Comments

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

Examples

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• OEIS Wiki, Index entries for number of divisors

Crossrefs

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

Programs

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

Mathematica

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

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

Extensions

Extended by T. D. Noe, May 08 2011