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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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9
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 (bowerc(AT)usa.net) May 18, 2005.
Number of primitive Pythagorean triangles with leg 4n. For smallest (even) leg of exactly 2^n PPTs, see A088860. - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 02]
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REFERENCES
| Henjin Chi and Raymond Killgrove, Problem 1447, Crux Math 15(5), May 1989 [Ant King, 15 Mar 2011].
Henjin Chi and Raymond Killgrove, Solution to Problem 1447, Crux Math 16(7), September 1990 [Ant King, 15 Mar 2011].
L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213. (See Table 1.)
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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)
Eric Weisstein's World of Mathematics, Unitary Divisor
Wikipedia, Unitary_divisor
N. J. A. Sloane, Transforms
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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 (blekraj(AT)yahoo.com), 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
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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
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MATHEMATICA
| a[n_] := Length[Select[Divisors[n], OddQ[ # ]&&GCD[ #, n/# ]==1&]]
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PROG
| (Haskell)
a068068 = length . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012
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CROSSREFS
| Cf. A056901, A068067.
Sequence in context: A201219 A080942 A099812 * A193523 A092505 A066086
Adjacent sequences: A068065 A068066 A068067 * A068069 A068070 A068071
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KEYWORD
| nonn,mult,changed
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 19 2002
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EXTENSIONS
| Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Jun 08 2002
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