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A226165
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Squarefree part of A077425(n) (numbers 4*k+1, k>=0, not a square).
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3
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5, 13, 17, 21, 29, 33, 37, 41, 5, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 13, 5, 129, 133, 137, 141, 145, 149, 17, 157, 161, 165, 173, 177, 181, 185, 21, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 5, 249, 253, 257, 29
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OFFSET
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1,1
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COMMENTS
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a(n) == 1 (mod 4), n >= 1. This is because 4*k+1, k>=0, not a square, can only have an even number of odd primes of the type 3 (mod 4) with odd exponents in the prime number factorization. The squarefree part of 4*k+1 has then an even number (maybe 0) of primes of the type 3 (mod 4). Examples:
a(4) = 21 = 3*7, a(6) = 33 = 3*11.
D(n) = A077425(n) are the 1 (mod 4) discriminants of indefinite binary quadratic forms (they are the odd numbers from A079896). sqrt(D(n)) becomes then, up to an integer factor, sqrt(a(n)), which defines a real quadratic number field Q(sqrt(a(n))) with a basis <1, omega(a(n))> for the ring of integers in this field, where omega(a(n)) = (1 + sqrt(a(n))/2. Example: sqrt(D(9)) = sqrt(45) = 3*sqrt(a(9)) = 3*sqrt(5), with omega(5) = (1 + sqrt(5))/2 (the golden section) for Q(sqrt(5)) = Q(omega(5)).
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LINKS
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FORMULA
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MATHEMATICA
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SquareFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]); SquareFreePart /@ (4*Range[65] + 1) // DeleteCases[#, 1] & (* Jean-François Alcover, Jun 14 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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