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 1-10 of 15 results. Next

A162688 Strictly positive numbers n such that 12*n/(12+n) are integers.

Original entry on oeis.org

4, 6, 12, 24, 36, 60, 132
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 12 row of A127730.
The ansatz 12*n/(12+n)=j (any integer j) yields n=12*j/(12-j) which demonstrates that the sequence is finite if n>=0. [R. J. Mathar, Jul 13 2009]

Crossrefs

Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=12;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]],AppendTo[lst,n]],{n,9!}];lst

Extensions

Edited by R. J. Mathar, Jul 13 2009

A162689 Strictly positive numbers n such that (18*n)/(18+n) are integers.

Original entry on oeis.org

9, 18, 36, 63, 90, 144, 306
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 18th row of A127730.
The ansatz 18*n/(18+n)=j (any integer j) yields n=18*j/(18-j) which demonstrates that the sequence is finite if n>=0. [R. J. Mathar, Jul 13 2009]

Crossrefs

Cf. A162688
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=18;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]],AppendTo[lst,n]],{n,9!}];lst
Select[Range[400],IntegerQ[(18#)/(18+#)]&] (* Harvey P. Dale, Nov 12 2011 *)

Extensions

Edited by R. J. Mathar, Jul 13 2009

A162690 Strictly positive numbers n such that 20*n/(20+n) are integers.

Original entry on oeis.org

5, 20, 30, 60, 80, 180, 380
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 20th row of A127730.
The ansatz 20*n/(20+n)=j (any integer j) yields n=20*j/(20-j) which demonstrates that the sequence is finite if n>=0. [R. J. Mathar, Jul 13 2009]

Crossrefs

Cf. A162688, A162689
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=20;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst

Extensions

Edited by R. J. Mathar, Jul 13 2009

A162691 Strictly positive numbers n such that 24*n/(24+n) is an integer.

Original entry on oeis.org

8, 12, 24, 40, 48, 72, 120, 168, 264, 552
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 24th row of A127730.
The ansatz 24*n/(24+n)=j (any integer j) yields n=24*j/(24-j) which demonstrates that the sequence is finite if n>=0. [R. J. Mathar, Jul 13 2009]

Crossrefs

Cf. A162688, A162689, A162690
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=24;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]],AppendTo[lst,n]],{n,9!}];lst
Select[Range[600],IntegerQ[(24#)/(24+#)]&] (* Harvey P. Dale, Jul 30 2021 *)

Extensions

Edited by R. J. Mathar, Jul 13 2009

A063427 a(n) is the smallest positive integer k such that n*k/(n+k) is an integer.

Original entry on oeis.org

2, 6, 4, 20, 3, 42, 8, 18, 10, 110, 4, 156, 14, 10, 16, 272, 9, 342, 5, 28, 22, 506, 8, 100, 26, 54, 21, 812, 6, 930, 32, 66, 34, 14, 12, 1332, 38, 78, 10, 1640, 7, 1806, 44, 30, 46, 2162, 16, 294, 50, 102, 52, 2756, 27, 66, 8, 114, 58, 3422, 12, 3660, 62, 18, 64, 104
Offset: 2

Views

Author

Henry Bottomley, Jul 19 2001

Keywords

Comments

This produces the smallest positive integer value for n*k/(n+k).
Equivalently, smallest c such that 1/n + 1/c = 1/b has integer solutions.
Largest c is 1/(n(n-1)) since 1/n + 1/(n(n-1)) = 1/(n-1).
Let L(x,y)=x+y be the "basic" linear form. Let Q(x,y) = x^2 + x*y + y^2 be the "basic" quadratic form. Let C(x,y) = x^3 + y^3 + x^2*y + x*y^2 + x*y + x^2 + y^2 + x + y be the "basic" cubic form. Then a(n) = min(x/Q(x,n)=0 mod L(x,n)) = min(x/C(x,n) = 0 mod L(x,n)). - Benoit Cloitre, Jan 02 2002
For p=prime, a(p^k) = p^k*(p-1). - Leroy Quet, Jan 25 2007
a(n) = n*(d(i)-d(i-1))/d(i-1), where d(i) is the i-th divisor of n that minimizes (d(i)-d(i-1))/d(i-1) with i>=2. In general, let f(n) be an integer function, then n*f(n)/(n+f(n))=c, c positive integer, has a solution only if f(n) >= n*(d(i)-d(i-1))/d(i-1). - Ctibor O. Zizka, Sep 17 2015

Examples

			a(6) = 3 because 6*3/(6+3)=2 is the smallest integer of the form 6*k/(6+k).
a(10) = 10 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. A063428, A127730.

Programs

  • Mathematica
    Table[k=1;While[!IntegerQ[(k n)/(k+n)],k++];k,{n,2,70}] (* Harvey P. Dale, Jun 24 2011 *)
  • PARI
    a(n) = { my(k=1); while (n*k%(n + k), k++); k } \\ Harry J. Smith, Aug 20 2009

Formula

a(n) = n*A063428(n)/(n-A063428(n)).

Extensions

New description from Benoit Cloitre, Dec 30 2001
Entry revised by N. J. A. Sloane, Feb 13 2007
Definition revised by Franklin T. Adams-Watters, Aug 07 2009

A162692 Strictly positive numbers n such that 28*n/(28+n) are integers.

Original entry on oeis.org

21, 28, 70, 84, 168, 364, 756
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 28th row of A127730.
The ansatz 28*n/(28+n)=j (any integer j) yields n=28*j/(28-j) which demonstrates that the sequence is finite if n>=0. [R. J. Mathar, Jul 13 2009]

Crossrefs

Cf. A162688, A162689, A162690, A162691
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=28;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]],AppendTo[lst,n]],{n,9!}];lst

Extensions

Edited by R. J. Mathar, Jul 13 2009

A162693 Strictly positive numbers n such that 30*n/(30+n) are integers.

Original entry on oeis.org

6, 15, 20, 30, 45, 60, 70, 120, 150, 195, 270, 420, 870
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 30th row of A127730.
The ansatz 30*n/(30+n)=j (any integer j) yields n=30*j/(30-j) which demonstrates that the sequence is finite if n>=0. [R. J. Mathar, Jul 13 2009]

Crossrefs

Cf. A162688, A162689, A162690, A162691, A162692
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=30;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]],AppendTo[lst,n]],{n,9!}];lst
Select[Range[1000],IntegerQ[(30#)/(30+#)]&] (* Harvey P. Dale, Sep 25 2019 *)

Extensions

Edited by R. J. Mathar, Jul 13 2009

A162694 Strictly positive numbers n such that 36*n/(36+n) are integers.

Original entry on oeis.org

12, 18, 36, 45, 72, 108, 126, 180, 288, 396, 612, 1260
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 10 2009

Keywords

Comments

The 36th row of A127730.
The ansatz 36*n/(36+n) = j (any integer j) yields n = 36*j/(36-j) which demonstrates that the sequence is finite if n >= 0. - R. J. Mathar, Jul 13 2009
Positive integers n such that half of the harmonic mean of 36 and n is an integer. - Wesley Ivan Hurt, Sep 07 2014

Crossrefs

Cf. A162688, A162689, A162690, A162691, A162692, A162693
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    Select[Range[1000], IntegerQ[36#/(36 + #)] &] (* Alonso del Arte, Sep 06 2014 *)

Extensions

Edited by R. J. Mathar, Jul 13 2009

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

A162817 Positive numbers n such that 40*n/(40+n) are integers.

Original entry on oeis.org

10, 24, 40, 60, 120, 160, 280, 360, 760, 1560
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 13 2009

Keywords

Crossrefs

Cf. A162688, A162689, A162690, A162691, A162692, A162693, A162694
Cf. A127730. [From Franklin T. Adams-Watters, Aug 07 2009]

Programs

  • Mathematica
    f[a_,b_]:=(a*b)/(a+b); a=40;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst
Select[Range[1600],IntegerQ[(40#)/(40+#)]&] (* Harvey P. Dale, Oct 11 2011 *)

Extensions

Keywords fini,full added by R. J. Mathar, Jul 31 2009
Showing 1-10 of 15 results. Next

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

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