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.

A137488 Numbers with 25 divisors.

Original entry on oeis.org

1296, 10000, 38416, 50625, 194481, 234256, 456976, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 6765201, 9150625, 10556001, 11316496, 14776336, 16777216, 17850625, 22667121, 29986576, 35153041, 45212176, 52200625
Offset: 1

Views

Author

R. J. Mathar, Apr 22 2008

Keywords

Comments

Maple implementation: see A030513.
Numbers of the form p^24 (24th powers of A000040, subset of A010812) or p^4*q^4 (A189991), where p and q are distinct primes. - R. J. Mathar, Mar 01 2010

Links

Crossrefs

Cf. A000005, A010812, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283, A135581, A175755.

Programs

  • Haskell
    a137488 n = a137488_list !! (n-1)
a137488_list = m (map (^ 24) a000040_list) (map (^ 4) 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
    lst = {}; Do[If[DivisorSigma[0, n] == 25, Print[n]; AppendTo[lst, n]], {n, 55000000}]; lst (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
Select[Range[5221*10^4],DivisorSigma[0,#]==25&] (* Harvey P. Dale, Mar 11 2019 *)
  • PARI
    is(n)=numdiv(n)==25 \\ Charles R Greathouse IV, Jun 19 2016
  • Python
    from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A137488(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 bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

A000005(a(n)) = 25.
Sum_{n>=1} 1/a(n) = (P(4)^2 - P(8))/2 + P(24) = 0.000933328..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

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

A201266 The seventh divisor of numbers with exactly 49 divisors.

Original entry on oeis.org

9, 16, 16, 27, 49, 22, 26, 81, 32, 125, 32, 81, 32, 81, 125, 81, 32, 32, 169, 81, 37, 343, 41, 289, 43, 87, 343, 93, 47, 361, 53, 111, 529, 59, 343, 61, 123, 129, 361, 64, 141, 64, 1331, 625, 64, 625, 64, 159, 529, 64, 177, 64, 183, 625, 1331, 64, 201, 64
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 29 2011

Keywords

Examples

			a(1) = A114334(7);
a(2) = A159765(7).

Links

Crossrefs

Cf. A175755, A135581.

Programs

  • Haskell
    a201266 n = [d | d <- [1..], a175755 n `mod` d == 0] !! 6
  • Python
    from math import isqrt
from sympy import primepi, integer_nthroot, primerange, divisors
def A201266(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 divisors(bisection(f,n,n))[6] # Chai Wah Wu, Feb 22 2025

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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