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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. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

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

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 ((A077610(n,k) mod 2): k = 1..A034444(k)) = sum ((A206778(n,k) mod 2): k = 1..A034444(k)). - Reinhard Zumkeller, Feb 12 2012

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.

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.

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} A008683(2d)^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

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 *)

PROG

(Haskell)

a068068 = length . filter odd . a077610_row

-- Reinhard Zumkeller, Feb 12 2012

(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

CROSSREFS

Cf. A056901, A068067, A008683.

Sequence in context: A080942 A099812 A246600 * A193523 A092505 A066086

Adjacent sequences:  A068065 A068066 A068067 * A068069 A068070 A068071

KEYWORD

nonn,mult,easy

AUTHOR

Robert G. Wilson v, Feb 19 2002

EXTENSIONS

Edited by Dean Hickerson, Jun 08 2002

STATUS

approved

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Last modified January 27 08:06 EST 2021. Contains 340460 sequences. (Running on oeis4.)