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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. 16
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.
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
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.
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} 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
a(n) = Sum_{d divides n, d odd} mu(d)^2. - Peter Bala, Feb 01 2024
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
Sequence in context: A080942 A099812 A246600 * A193523 A092505 A066086
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 May 19 05:06 EDT 2024. Contains 372666 sequences. (Running on oeis4.)