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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.

%I A068068 #106 Jul 21 2025 06:52:18
%S A068068 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,
%T A068068 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,
%U A068068 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
%N 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.
%C A068068 Shadow transform of triangular numbers.
%C A068068 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.
%C A068068 Number of primitive Pythagorean triangles with leg 4n. For smallest (even) leg of exactly 2^n PPTs, see A088860. - _Lekraj Beedassy_, Jul 12 2006
%C A068068 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
%C A068068 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
%C A068068 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
%C A068068 a(n) is also the number of even unitary divisors of 2*n. - _Amiram Eldar_, Jan 28 2023
%H A068068 Reinhard Zumkeller, <a href="/A068068/b068068.txt">Table of n, a(n) for n = 1..10000</a>
%H A068068 Henjin Chi and Raymond Killgrove, <a href="https://cms.math.ca/crux/backfile/Crux_v15n05_May.pdf">Problem 1447</a>, Crux Math 15(5), May 1989.
%H A068068 Henjin Chi and Raymond Killgrove, <a href="https://cms.math.ca/crux/backfile/Crux_v16n07_Sep.pdf">Solution to Problem 1447</a>, Crux Math 16(7), September 1990.
%H A068068 L. J. Gerstein, <a href="http://www.jstor.org/stable/30044157">Pythagorean triples and inner products</a>, Math. Mag. 78 (2005), 205-213.
%H A068068 Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150. (<a href="http://user.math.uzh.ch/halbeisen/publications/psf/seq.ps">ps</a>, <a href="http://user.math.uzh.ch/halbeisen/publications/pdf/seq.pdf">pdf</a>); see Definition 7 for the shadow transform.
%H A068068 Lorenz Halbeisen, <a href="http://www.iam.fmph.uniba.sk/amuc/_vol-74/_no_2/_halbeisen/halbeisen.html">A number-theoretic conjecture and its implication for set theory</a>, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
%H A068068 R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet series of multiplicative arithmetic functions</a>, arXiv:1106.4038 [math.NT], 2011-2012.
%H A068068 OEIS Wiki, <a href="https://oeis.org/wiki/Shadow_transform">Shadow transform</a>.
%H A068068 Neville Robbins, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/44-4/quartRobbins04_2006.pdf">On the number of primitive Pythagorean triangles with a given inradius</a>, Fibonacci Quart. 44(4) (2006), 368-369.
%H A068068 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%H A068068 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.
%H A068068 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.
%F A068068 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.
%F A068068 Multiplicative with a(2^e) = 1, a(p^e) = 2, p>2. - _Christian G. Bower_ May 18 2005
%F A068068 a(n) = A024361(4n). - _Lekraj Beedassy_, Jul 12 2006
%F A068068 Dirichlet g.f.: zeta^2(s)/ ( zeta(2*s)*(1+2^(-s)) ). Dirichlet convolution of A034444 and A154269. - _R. J. Mathar_, Apr 16 2011
%F A068068 a(n) = Sum_{d|n} mu(2*d)^2. - _Ridouane Oudra_, Aug 11 2019
%F A068068 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
%F A068068 a(n) = Sum_{d divides n, d odd} mu(d)^2. - _Peter Bala_, Feb 01 2024
%p A068068 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
%t A068068 a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]]
%t A068068 a[n_] := 2^(PrimeNu[n]+Mod[n, 2]-1); Array[a, 105] (* _Jean-François Alcover_, Dec 01 2015 *)
%t A068068 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 *)
%o A068068 (Haskell)
%o A068068 a068068 = length . filter odd . a077610_row
%o A068068 -- _Reinhard Zumkeller_, Feb 12 2012
%o A068068 (PARI) a(n) = sumdiv(n, d, (d%2)*(gcd(d, n/d)==1)); \\ _Michel Marcus_, May 13 2014
%o A068068 (PARI) a(n) = 2^omega(n>>valuation(n,2)) \\ _Charles R Greathouse IV_, May 14 2014
%Y A068068 Cf. A001221, A001620, A024361, A034444, A056901, A068067, A008683, A323239.
%K A068068 nonn,mult,easy
%O A068068 1,3
%A A068068 _Robert G. Wilson v_, Feb 19 2002
%E A068068 Edited by _Dean Hickerson_, Jun 08 2002