login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A018892 Number of ways to write 1/n as a sum of exactly 2 unit fractions. 24
1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 14, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 14, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 23, 2, 5, 8, 7, 5, 14, 2, 8, 5, 14, 2, 18, 2, 5, 8, 8, 5, 14, 2, 14, 5, 5, 2, 23, 5, 5, 5, 11, 2, 23, 5, 8, 5, 5, 5, 17, 2, 8, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = (tau(n^2)+1)/2. Number of elements in the set {(x,y): x|n, y|n, x<=y, GCD(x,y)=1}. Number of divisors of n^2 less than or equal to n. - Vladeta Jovovic, May 03 2002

Equivalently number of pairs (x,y) such that LCM(x,y)=n. - Benoit Cloitre, May 16 2002

Number of right triangles with an integer hypotenuse and height n. - Reinhard Zumkeller, Jul 10 2002

The triangles are to be considered as resting on their hypotenuse, with the height measured to the right angle. - Franklin T. Adams-Watters, Feb 19 2015

Except for the initial term, each entry is at least equal to 2 because of the identities 1/n = 1/(2*n) + 1/2n = 1/(n+1) + 1/(n*(n+1)). - Lekraj Beedassy, May 04 2004

REFERENCES

K. S. Brown, Posting to netnews group sci.math, Aug 17 1996.

L. E. Dickson, History of The Theory of Numbers, Vol. 2 p. 690, Chelsea NY 1923.

A. M. & I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Vol. 1 pp. 8;60 Prob. 19 Dover NY

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892

Roger B. Eggleton, Problem 10501(a), American Mathematical Monthly, Vol. 105, No. 4, 1998 p. 372.

Project Euler, Problem 379

FORMULA

If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1 + 1)(2*a2 + 1) ... (2*at + 1) + 1)/2.

a(n) = A063647(n)+1 = A046079(2n)+1. - Lekraj Beedassy, Dec 01 2003

a(n) = Sum(d|n, phi(2^omega(d)), where phi is A000010 and omega is A001221. - Enrique Pérez Herrero, Apr 13 2012

a(n) = A000005(n) + A089233(n). - James Spahlinger, Feb 16 2016

EXAMPLE

Examples:

n=1: 1/1 = 1/2 + 1/2.

n=2: 1/2 = 1/4 + 1/4 = 1/3 + 1/6.

n=3: 1/3 = 1/6 + 1/6 = 1/4 + 1/12.

MATHEMATICA

f[j_, n_] := (Times @@ (j(Last /@ FactorInteger[n]) + 1) + j - 1)/j; Table[f[2, n], {n, 96}] (*Robert G. Wilson v, Aug 03 2005 *)

a[n_] := (DivisorSigma[0, n^2] + 1)/2; Table[a[n], {n, 1, 99}](* Jean-François Alcover, Dec 19 2011, after Vladeta Jovovic *)

PROG

(PARI) A018892(n)=(numdiv(n^2)+1)/2 \\ M. F. Hasler, Dec 30 2007

(PARI) A018892s(n)=local(t=divisors(n^2)); vector((#t+1)/2, i, [n+t[i], n+n^2/t[i]]) /* show solutions */ \\ M. F. Hasler, Dec 30 2007

(PARI) a(n)=sumdiv(n, d, sum(i=1, d, lcm(d, i)==n)) \\ Charles R Greathouse IV, Apr 08 2012

(Haskell)

a018892 n = length [d | d <- [1..n], n^2 `mod` d == 0]

-- Reinhard Zumkeller, Jan 08 2012

CROSSREFS

Records: A126097, A126098. Cf. A048691, A063647.

Sequence in context: A160273 A141822 A033099 * A100565 A244098 A285573

Adjacent sequences:  A018889 A018890 A018891 * A018893 A018894 A018895

KEYWORD

nonn,easy,nice

AUTHOR

Robert G. Wilson v

EXTENSIONS

More terms from David W. Wilson, Sep 15 1996

First example corrected by Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 02 2009

Incorrect Mathematica program deleted by N. J. A. Sloane, Jul 08 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 17 07:13 EDT 2017. Contains 290635 sequences.