cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A114334 Divisors of 6^6.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 144, 162, 192, 216, 243, 288, 324, 432, 486, 576, 648, 729, 864, 972, 1296, 1458, 1728, 1944, 2592, 2916, 3888, 5184, 5832, 7776, 11664, 15552, 23328, 46656
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 07 2006

Keywords

Comments

Subsequence of A003586; 128 = 2^(6+1) is the smallest 3-smooth number not dividing 6^6.
a(49) = A175755(1) = 46656 = smallest number with exactly 49 divisors; a(7) = A201266(1). - Reinhard Zumkeller, Nov 29 2011

Links

Crossrefs

A291713 lists terms a(14)-a(22).

Programs

Formula

a(n) = A027750(46656,n) for n = 1 .. A000005(46656). - Reinhard Zumkeller, Jan 07 2014

A135581 The 5th divisor of numbers with 25 divisors.

Original entry on oeis.org

6, 8, 8, 15, 21, 11, 13, 27, 16, 35, 16, 27, 16, 27, 55, 27, 16, 16, 16, 65, 27, 16, 77, 16, 85, 16, 29, 91, 31, 16, 95, 16, 37, 115, 16, 119, 16, 41, 43, 133, 16, 47, 16, 143, 125, 16, 125, 16, 53, 161, 16, 59, 16, 61, 125, 187, 16, 67, 16, 203, 125, 16, 209, 71, 16, 125
Offset: 1

Views

Author

G. H. Ens (GerardEns(AT)gmail.com), Feb 24 2008

Keywords

Comments

n=1 means the first number that has 25 divisors (1296), 6 is the 5th divisor of 1296. The second number with 25 divisors is 10000 and its 5th divisor is 8
This is one example of such a sequence where the divisor index is the square root of the total number of divisors (self included).
Other examples would be the 6th divisor of numbers with 36 divisors, 7th divisor of numbers with 49 divisors, etc.
Choice of the square root is arbitrary.
All but 16 primes {2, 3, 5, 7, 17, 19, 23, 83, 89, 97, 101, 103, 107, 109, 113} are in this sequence; p^3 and p^4 are in this sequence for all prime p; pq is in this sequence for all prime p and q with p < q < p^2. No other terms are members. - Charles R Greathouse IV, Nov 28 2011

Examples

			a(1) = 6 because 6 is the 5th divisor of 1296 and 1296 is the first number with 25 divisors.
a(2) = 8 because 8 is the 5th divisor of 10000 and 10000 is the second number with 25 divisors.

Links

Crossrefs

Cf. A005179, A137488, A200722, A201266.

Programs

  • Haskell
    a135581 n = [d | d <- [1..], a137488 n `mod` d == 0] !! 4
-- Reinhard Zumkeller, Nov 29 2011
  • Mathematica
    upto=10^10;With[{max1=Ceiling[Power[upto, (4)^-1]],max2=Ceiling[ Power[ upto, (24)^-1]]},Take[Divisors[#][[5]]&/@Select[Union[Join[ Range[ max2]^24, Times@@@(Subsets[Range[max1],{2}]^4)]],DivisorSigma[0,#] == 25&], Ceiling[max1/4]]] (* Harvey P. Dale, Nov 25 2011 *)
  • Python
    from math import isqrt
from sympy import primepi, integer_nthroot, primerange, divisors
def A135581(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,4)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)))-primepi(integer_nthroot(x,24)[0])
    return divisors(bisection(f,n,n))[4] # Chai Wah Wu, Feb 22 2025

Extensions

Corrected and extended by R. J. Mathar, Apr 21 2008. The original entries were wrong from the 16th term onwards.

A175755 Numbers with 49 divisors.

Original entry on oeis.org

46656, 1000000, 7529536, 11390625, 85766121, 113379904, 308915776, 1291467969, 1544804416, 1838265625, 3010936384, 3518743761, 9474296896, 17596287801, 27680640625, 34296447249, 38068692544, 56800235584, 75418890625, 107918163081, 164206490176, 208422380089
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

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

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

A159765 Divisors of 1000000.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 800, 1000, 1250, 1600, 2000, 2500, 3125, 4000, 5000, 6250, 8000, 10000, 12500, 15625, 20000, 25000, 31250, 40000, 50000, 62500, 100000, 125000, 200000, 250000, 500000, 1000000
Offset: 1

Views

Author

Zerinvary Lajos, Apr 21 2009

Keywords

Comments

a(49) = A175755(2); a(7) = A201266(2). - Reinhard Zumkeller, Nov 29 2011

Links

Programs

  • Haskell
    a159765 n = a159765_list !! (n-1)
a159765_list = a027750_row 1000000  -- Reinhard Zumkeller, Jan 07 2014
  • Mathematica
    Divisors[10^6] (* Paolo Xausa, Jul 01 2024 *)
  • PARI
    divisors(10^6) \\ Charles R Greathouse IV, Jun 21 2017
  • Sage
    a = 100000; a.divisors()

Formula

a(n) = A027750(1000000,n) for n = 1 .. A000005(1000000). - Reinhard Zumkeller, Jan 07 2014
Showing 1-4 of 4 results.

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