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A091285
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Numbers k such that sigma_3(k) is divisible by the square of phi(k).
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7
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1, 2, 3, 6, 14, 42, 3810, 318990, 13243560, 1108809240, 1719507048, 25330080090, 271984504290
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OFFSET
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1,2
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COMMENTS
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The first 8 terms are solutions to: {sigma_{6j+3}(x)/phi(x)^2 is integer, for j=1,...,300}. A proof is possible with knowledge of respective divisors of sigma_k(x) and phi(x).
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LINKS
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EXAMPLE
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k = 14: phi(k)^2 = 36, sigma_3(k) = 3096 = 36*86.
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MATHEMATICA
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Empirical test for very high powers of divisors is: t = {1, 2, 3, 6, 14, 42, 3810, 13243560} Table[{6*j+3, Union[Table[IntegerQ[DivisorSigma[6*j + 3, Part[t, k]]/EulerPhi[Part[t, k]]^2], {k, 1, 8}]]}, {j, 1, 300}]; output={exponent, True}.
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PROG
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(PARI) for(n = 1, 10^9, if(sigma(n, 3) % (eulerphi(n)^2) == 0, print1(n, ", "))) \\ Ryan Propper, Jan 18 2008
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CROSSREFS
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KEYWORD
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more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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