A068068 Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 1, 4, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 2, 4, 2, 2, 4, 1, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 4, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 4, 4, 2, 4, 2, 4, 2, 2, 2, 4, 2, 2, 4, 2, 2, 8

Offset: 1

Author

Robert G. Wilson v, Feb 19 2002

Keywords

• nonn
• mult
• easy

Comments

Shadow transform of triangular numbers.

a(n) is the number of primitive Pythagorean triangles with inradius n. For the smallest inradius of exactly 2^n primitive Pythagorean triangles see A070826.

Number of primitive Pythagorean triangles with leg 4n. For smallest (even) leg of exactly 2^n PPTs, see A088860. - Lekraj Beedassy, Jul 12 2006

As shown by Chi and Killgrove, a(n) is the total number of primitive Pythagorean triples satisfying area = n * perimeter, or equivalently 2 raised to the power of the number of distinct, odd primes contained in n. - Ant King, Mar 15 2011

This is the case k=0 of the sum over the k-th powers of the odd unitary divisors of n, which is multiplicative with a(2^e)=1 and a(p^e)=1+p^(e*k), p>2, and has Dirichlet g.f. zeta(s)*zeta(s-k)*(1-2^(k-s))/( zeta(2s-k)*(1-2^(k-2*s)) ). - R. J. Mathar, Jun 20 2011

Also the number of odd squarefree divisors of n: a(n) = Sum_{k = 1..A034444(k)} (A077610(n,k) mod 2) = Sum_{k = 1..A034444(k)} (A206778(n,k) mod 2). - Reinhard Zumkeller, Feb 12 2012

a(n) is also the number of even unitary divisors of 2*n. - Amiram Eldar, Jan 28 2023

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Henjin Chi and Raymond Killgrove, Problem 1447, Crux Math 15(5), May 1989.
• Henjin Chi and Raymond Killgrove, Solution to Problem 1447, Crux Math 16(7), September 1990.
• L. J. Gerstein, Pythagorean triples and inner products, Math. Mag. 78 (2005), 205-213.
• Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
• Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
• R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012.
• OEIS Wiki, Shadow transform.
• Neville Robbins, On the number of primitive Pythagorean triangles with a given inradius, Fibonacci Quart. 44(4) (2006), 368-369.
• N. J. A. Sloane, Transforms.
• Eric Weisstein's World of Mathematics, Unitary Divisor.
• Wikipedia, Unitary divisor.

Crossrefs

Cf. A001221, A001620, A024361, A034444, A056901, A068067, A008683, A323239.

Programs

Haskell

a068068 = length . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

Maple

A068068 := proc(n) local a,f; a :=1 ; for f in ifactors(n)[2] do if op(1,f) > 2 then a := a*2 ; end if; end do: a ; end proc: # R. J. Mathar, Apr 16 2011

Mathematica

a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]]
a[n_] := 2^(PrimeNu[n]+Mod[n, 2]-1); Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)
f[p_, e_] := If[p == 2, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

PARI

a(n) = sumdiv(n, d, (d%2)*(gcd(d, n/d)==1)); \\ Michel Marcus, May 13 2014

PARI

a(n) = 2^omega(n>>valuation(n,2)) \\ Charles R Greathouse IV, May 14 2014

Formula

a(n) = A034444(2n)/2. If n is even, a(n) = 2^(omega(n)-1); if n is odd, a(n) = 2^omega(n). Here omega(n) = A001221(n) is the number of distinct prime divisors of n.

Multiplicative with a(2^e) = 1, a(p^e) = 2, p>2. - Christian G. Bower May 18 2005

a(n) = A024361(4n). - Lekraj Beedassy, Jul 12 2006

Dirichlet g.f.: zeta^2(s)/ ( zeta(2*s)*(1+2^(-s)) ). Dirichlet convolution of A034444 and A154269. - R. J. Mathar, Apr 16 2011

a(n) = Sum_{d|n} mu(2*d)^2. - Ridouane Oudra, Aug 11 2019

Sum_{k=1..n} a(k) ~ 4*n*((log(n) + 2*gamma - 1 + log(2)/3) / Pi^2 - 12*zeta'(2) / Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 18 2020

a(n) = Sum_{d divides n, d odd} mu(d)^2. - Peter Bala, Feb 01 2024

Extensions

Edited by Dean Hickerson, Jun 08 2002