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A068068
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Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d,n/d)=1.
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16
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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
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OFFSET
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1,3
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COMMENTS
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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
a(n) is also the number of even unitary divisors of 2*n. - Amiram Eldar, Jan 28 2023
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LINKS
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Henjin Chi and Raymond Killgrove, Problem 1447, Crux Math 15(5), May 1989.
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.
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FORMULA
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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.
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
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MAPLE
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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
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MATHEMATICA
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a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]]
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 *)
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PROG
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(Haskell)
a068068 = length . filter odd . a077610_row
(PARI) a(n) = sumdiv(n, d, (d%2)*(gcd(d, n/d)==1)); \\ Michel Marcus, May 13 2014
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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