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.

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

A121067 a(n) = the n-th divisor of n^n (when the positive divisors of n^n are written in order from smallest to largest).

Original entry on oeis.org

1, 2, 9, 8, 625, 8, 117649, 128, 6561, 32, 25937424601, 27, 23298085122481, 112, 375, 32768, 48661191875666868481, 72, 104127350297911241532841, 250, 2401, 1024, 907846434775996175406740561329, 162, 59604644775390625, 2704, 2541865828329
Offset: 1

Views

Author

Leroy Quet, Aug 10 2006

Keywords

Comments

This is also the n-th divisor of n^(n-1); any divisor with a factor of p^n is preceded by n smaller powers of p in the divisor list. [Franklin T. Adams-Watters, Sep 21 2009]

Examples

			1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64,... is the beginning of the sequence of divisors of 6^6 = 46656. 8 is the 6th term of this sequence of divisors (which is sequence A114334), so a(6) = 8.

Links

Crossrefs

Cf. A000312.
Cf. A000169. [Franklin T. Adams-Watters, Sep 21 2009]

Programs

  • GAP
    List([1..30],n->DivisorsInt(n^n)[n]); # Muniru A Asiru, Mar 06 2019
  • Maple
    a:= n-> sort([numtheory[divisors](n^(n-1))[]])[n]:
seq(a(n), n=1..30);  # Alois P. Heinz, Oct 09 2016
  • Mathematica
    Table[Divisors[n^n][[n]], {n, 27}] (* Michael De Vlieger, Sep 19 2017 *)
  • PARI
    m=27;for(n=1,m,d=divisors(n^n);print1(d[n],",")) \\ Klaus Brockhaus, Aug 14 2006

Formula

a(n) <= A020639(n)^n, with equality for n a prime power. - Charlie Neder, Mar 06 2019

Extensions

More terms from Joshua Zucker, Klaus Brockhaus and Jason Earls, Aug 11 2006

A291713 144 * preferred ratios of room width / length in Alberti's Ten Books on Architecture.

Original entry on oeis.org

36, 48, 54, 64, 72, 81, 96, 108, 144
Offset: 1

Views

Author

Hugo Pfoertner, Aug 30 2017

Keywords

Comments

Coincides with A114334(14) ... A114334(22) (Divisors of 6^6).

Examples

			The following table is provided in the Zuk article:
.
Table 38.9 Proportions derived from Alberti's preferred ratios
  Plan  Length      Mean              L : H : W
  ratio      Width             Height
   1:1    1    1    Any mean     1    1 : 1 : 1
   3:2    6    4    Arithmetic   5    6 : 5 : 4
   3:2   15   10    Harmonic    12   15 :12 :10
   4:3    8    6    Arithmetic   7    8 : 7 : 6
   2:1    4    2    Arithmetic   3    4 : 3 : 2
   9:4    9    4    Geometric    6    9 : 6 : 4
  16:9   16    9    Geometric   12   16 :12 : 9
   3:1    3    1    Arithmetic   2    3 : 2 : 1
   8:3    8    3    Fibonacci    5    8 : 5 : 3
   4:1    4    1    Geometric    2    4 : 2 : 1
.
a(8)=108 is in the sequence, because the ratio 144/108 corresponds to the plan ratio of 4:3.

Links

Crossrefs

Cf. A114334, A291714, A291715, A291719.
Showing 1-4 of 4 results.

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