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.

A063428 a(n) is the smallest positive integer of the form n*k/(n+k).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 6, 5, 10, 3, 12, 7, 6, 8, 16, 6, 18, 4, 12, 11, 22, 6, 20, 13, 18, 12, 28, 5, 30, 16, 22, 17, 10, 9, 36, 19, 26, 8, 40, 6, 42, 22, 18, 23, 46, 12, 42, 25, 34, 26, 52, 18, 30, 7, 38, 29, 58, 10, 60, 31, 14, 32, 40, 22, 66, 34, 46, 20, 70, 8, 72, 37, 30, 38, 28, 26
Offset: 2

Views

Author

Henry Bottomley, Jul 19 2001

Keywords

Comments

Or, smallest b such that 1/n + 1/c = 1/b has integer solutions.
Largest b is (n-1) since 1/n + 1/(n(n-1)) = 1/(n-1).
a(n) = smallest k such that k*n/(k-n) is an integer. - Derek Orr, May 29 2014

Examples

			a(6) = 2 because 6*3/(6+3) = 2 is the smallest integer of the form 6*k/(6+k).
a(10) = 5 since 1/10 + 1/10 = 1/5, 1/10 + 1/15 = 1/6, 1/10 + 1/40 = 1/8, 1/10 + 1/90 = 1/9 and so the first sum provides the value.

Links

Crossrefs

Cf. A018892, A063427, A127730, A063647, A063648, A063649, A066092.

Programs

  • Mathematica
    spi[n_]:=Module[{k=1},While[!IntegerQ[(n*k)/(n+k)],k++];(n*k)/(n+k)]; Array[ spi,80,2] (* Harvey P. Dale, May 05 2022 *)
  • PARI
    a(n)={my(k=1); if(n>1, while (n*k%(n + k), k++); n*k/(n + k))} \\ Harry J. Smith, Aug 20 2009

Formula

a(n) = n*A063427(n)/(n + A063427(n)) = 2n - A063649(n).
If n is prime a(n) = n - 1. - Benoit Cloitre, Dec 31 2001

Extensions

New description from Benoit Cloitre, Dec 31 2001
Entry revised by N. J. A. Sloane, Feb 13 2007
Definition revised by Franklin T. Adams-Watters, Aug 07 2009

A063649 Largest b such that 1/n=1/c+1/b has integer solutions with c>b.

Original entry on oeis.org

3, 4, 6, 6, 10, 8, 12, 12, 15, 12, 21, 14, 21, 24, 24, 18, 30, 20, 36, 30, 33, 24, 42, 30, 39, 36, 44, 30, 55, 32, 48, 44, 51, 60, 63, 38, 57, 52, 72, 42, 78, 44, 66, 72, 69, 48, 84, 56, 75, 68, 78, 54, 90, 80, 105, 76, 87, 60, 110, 62, 93, 112, 96, 90, 110, 68, 102, 92, 120
Offset: 2

Views

Author

Henry Bottomley, Jul 23 2001

Keywords

Comments

Smallest b is (n+1) since 1/n = 1/(n(n+1))+1/(n+1).

Examples

			a(10)=15 since 1/10=1/20+1/20=1/30+1/15=1/35+1/14=1/60+1/12=1/110+1/11, but the first sum does not have c>b, leaving the second sum to provide the value.

Links

Crossrefs

Cf. A018892, A063427, A063428, A063647, A063648.

Programs

  • Maple
    f:= proc(n) local b;
  for b from 2*n-1 by -1 do
     if n*b mod (b-n) = 0 then return b fi
od
end proc:
map(f, [$2..100]); # Robert Israel, Dec 01 2019
  • Mathematica
    a[n_] := n + SelectFirst[Divisors[n^2] // Reverse, #Jean-François Alcover, Jun 07 2020 *)

Formula

a(n) = n*A063648(n)/(A063648(n)-n) = 2n-A063428(n).
From Robert Israel, Dec 01 2019: (Start)
a(n) = n + A063717(n).
a(n) = n + 1 if and only if n is prime. (End)

A063718 a(n) is the smallest divisor of n^2 that is greater than n.

Original entry on oeis.org

4, 9, 8, 25, 9, 49, 16, 27, 20, 121, 16, 169, 28, 25, 32, 289, 27, 361, 25, 49, 44, 529, 32, 125, 52, 81, 49, 841, 36, 961, 64, 99, 68, 49, 48, 1369, 76, 117, 50, 1681, 49, 1849, 88, 75, 92, 2209, 64, 343, 100, 153, 104, 2809, 81, 121, 64, 171, 116, 3481, 72, 3721, 124
Offset: 2

Views

Author

Vladeta Jovovic, Aug 12 2001

Keywords

Comments

Larger of two distinct numbers with minimum sum whose geometric mean is n. E.g., a(12) = 16 as 12^2 = 144 = 1*144 = 2*72 = 3*48 = 4*36 = 6*24 = 8*18 = 9*16, etc. - Amarnath Murthy, Feb 15 2003

Examples

			a(45)=75 because divisors of 45^2 are {1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025} and the smallest element greater than 45 is 75.

Links

Crossrefs

A063648(n) = n + a(n), A063717(n) = n^2/A063718(n), A063427(n) = n - a(n).
Cf. A063717.

Programs

  • Maple
    with(numtheory): for n from 2 to 200 do a := divisors(n^2): b := a[(tau(n^2)-1)/2]: printf(`%d,`,n^2/b); od:
  • Mathematica
    sdgn[n_]:=Select[Divisors[n^2],#>n&,1]; Flatten[Array[sdgn,70]] (* Harvey P. Dale, Jun 18 2012 *)
  • PARI
    { for (n=2, 1000, d=divisors(n^2); write("b063718.txt", n, " ", d[2 + length(d)\2]) ) } \\ Harry J. Smith, Aug 28 2009
Showing 1-3 of 3 results.

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

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