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 1-3 of 3 results.

A018266 Divisors of 60.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Sequence is finite with last term a(12) = 60; A000005(60) = 12. - Reinhard Zumkeller, Dec 08 2009.
60 is a highly composite number: A002182(9) = 60. - Reinhard Zumkeller, Jun 21 2010
There are 35 ways to partition 60 as a sum of its distinct divisors (see A033630). This is more than any smaller number (hence 60 is listed in A065218). - Alonso del Arte, Oct 12 2017

Links

Crossrefs

Cf. A035303, A171425. - Reinhard Zumkeller, Dec 08 2009
Cf. A018253, A018256, A018261, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864. - Reinhard Zumkeller, Jun 21 2010

Programs

Formula

a(n) = n + floor((n-1)/6)*(60/(13-n)-n). - Aaron J Grech, Aug 11 2024

A055205 Number of nonsquare divisors of n^2.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 2, 5, 1, 9, 1, 5, 5, 4, 1, 9, 1, 9, 5, 5, 1, 13, 2, 5, 3, 9, 1, 19, 1, 5, 5, 5, 5, 16, 1, 5, 5, 13, 1, 19, 1, 9, 9, 5, 1, 17, 2, 9, 5, 9, 1, 13, 5, 13, 5, 5, 1, 33, 1, 5, 9, 6, 5, 19, 1, 9, 5, 19, 1, 23, 1, 5, 9, 9, 5, 19, 1, 17, 4, 5, 1, 33, 5, 5, 5, 13, 1, 33, 5, 9, 5, 5, 5
Offset: 1

Views

Author

Labos Elemer, Jun 19 2000

Keywords

Comments

Seems to be equal to the number of unordered pairs of coprime divisors of n. (Checked up to 2*10^14.) - Charles R Greathouse IV, May 03 2013
Outline of a proof for this observation, R. J. Mathar, May 05 2013: (Start)
i) To construct the divisors of n, write n=product_i p_i^e_i as the standard prime power decomposition, take any subset of the primes p_i (including the empty set representing the 1) and run with the associated list exponents from 0 up to their individual e_i.
To construct the *nonsquare* divisors of n, ensure that one or more of the associated exponents is/are odd. (The empty set is interpreted as 1^0 with even exponent.) To construct the nonsquare divisors of n^2, the principle remains the same, although the exponents may individually range from 0 up to 2*e_i.
The nonsquare divisor is therefore a nonempty product of prime powers (at least one) with odd exponents times a (potentially empty) product of prime powers (of different primes) with even exponents.
The nonsquare divisors of n^2 have exponents from 0 up to 2*e_i, but the subset of exponents in the "even/square" factor has e_i candidates (range 2, 4, .., 2*e_i) and in the "odd/nonsquare" factor also only e_i candidates (range 1,3,5,2*e_i-1).
ii) To construct the pairs of coprime divisors of n, take any two non-intersecting subsets of the set of p_i (possibly the empty subset which represents the factor 1), and let the exponents run from 1 up to their individual e_i in each of the two products.
iii) The bijection between the sets constructed in i) and ii) is given by mapping the two non-intersection prime sets onto each other, and observing that the numbers of compositions of exponents have the same orders in both cases.
(End)

Examples

			n = 8, d(64) = 7 and from the 7 divisors {1,4,16,64} are square and the remaining 3 = a(8).
n = 12, d(144) = 15, from which 6 divisors are squares {1,4,9,16,36,144} so a(12) = d(144)-d(12) = 9
a(60) = (number of terms of finite A171425) = 33. [_Reinhard Zumkeller_, Dec 08 2009]

Links

Crossrefs

Cf. A000005, A000290, A048691, A056595.
Cf. A001620, A013661, A082633, A201994, A306016.

Programs

  • Haskell
    a055205 n = length [d | d <- [1..n^2], n^2 `mod` d == 0, a010052 d == 0]
-- Reinhard Zumkeller, Aug 15 2011
  • Mathematica
    Table[Count[Divisors[n^2], d_ /;  ! IntegerQ[Sqrt[d]]], {n, 1, 95}] (* Jean-François Alcover, Mar 22 2011 *)
Table[DivisorSigma[0,n^2]-DivisorSigma[0,n],{n,100}] (* Harvey P. Dale, Sep 02 2017 *)
  • PARI
    a(n)=my(f=factor(n)[,2]);prod(i=1,#f,2*f[i]+1)-prod(i=1,#f,f[i]+1) \\ Charles R Greathouse IV, May 02 2013

Formula

a(n) = A000005(n^2)-A000005(n) because the number of square divisors of n^2 equals the number of divisors of n.
a(n) = A056595(A000290(n)).
a(n) = A048691(n) - A000005(n). - Reinhard Zumkeller, Dec 08 2009
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n)^2/2 + c_1 * log(n) + c_2), where c_1 = 3*gamma - 2*zeta'(2)/zeta(2) - zeta(2) - 1 = 0.226634..., c_2 = 3*gamma^2 - (2*gamma - 1)*zeta(2) - 3*gamma_1 + (1 - 3*gamma)*(2*zeta'(2)/zeta(2) + 1) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2) = -0.0529271..., gamma is Euler's constant (A001620), and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Dec 01 2023

A035303 Divisors of 3600.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100, 120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Sequence is finite with last term a(45)=3600=60^2; A048691(60)=45. [From Reinhard Zumkeller, Dec 08 2009]

Links

Crossrefs

Cf. A018266, A171425. [From Reinhard Zumkeller, Dec 08 2009]

Programs

  • Mathematica
    Divisors[ 3600 ]
  • PARI
    divisors(3600)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.