

A018892


Number of ways to write 1/n as a sum of exactly 2 unit fractions.


29



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
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OFFSET

1,2


COMMENTS

a(n) = (tau(n^2)+1)/2. Number of elements in the set {(x,y): xn, yn, 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
Also, 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. AdamsWatters, Feb 19 2015
a(n) >= 2 for n>=2 because of the identities 1/n = 1/(2*n) + 1/(2*n) = 1/(n+1) + 1/(n*(n+1)).  Lekraj Beedassy, May 04 2004
a(n) is the number of divisors of n^2 that are <= n; e.g., a(12) counts these 8 divisors of 12: 1,2,3,4,6,8,9,12.  Clark Kimberling, Apr 21 2019


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: Least common multiple count


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_{dn} 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}](* JeanFrançois 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: A033099 A330833 A337331 * 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



