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

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

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

A200722 Numbers that are the 5th-smallest divisor of n for some n with precisely 25 divisors.

Original entry on oeis.org

6, 8, 11, 13, 15, 16, 21, 27, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 81, 85, 91, 95, 115, 119, 125, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 221, 223
Offset: 1

Views

Author

Charles R Greathouse IV, Nov 29 2011

Keywords

Comments

Members of A135581, sorted. Characterizations (see Formulas section, below) are possible for similar sequences as well.

Links

Crossrefs

Cf. A135581.

Programs

  • PARI
    is(n)=my(f=factor(n));if(#f[,1]==1,f[1,2]==3||f[1,2]==4||(f[1,2]==1&&(f[1,1]>126||(f[1,1]<80&&f[1,1]>28)||f[1,1]==11||f[1,1]==13)),#f[,1]==2&&f[1,2]==1&&f[2,2]==1&&f[2,1]

Formula

Characterization: all terms of this sequence are of the form p, p^3, p^4, or pq where p and q are distinct primes. All but 16 primes {2, 3, 5, 7, 17, 19, 23, 83, 89, 97, 101, 103, 107, 109, 113} are in this sequence; all p^3 and p^4 are in this sequence; pq is in this sequence for all p < q < p^2.
Showing 1-4 of 4 results.

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