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A189000
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Bi-unitary multiperfect numbers.
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5
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1, 6, 60, 90, 120, 672, 2160, 10080, 22848, 30240, 342720, 523776, 1028160, 1528800, 6168960, 7856640, 7983360, 14443520, 22932000, 23569920, 43330560, 44553600, 51979200, 57657600, 68796000, 133660800, 172972800, 779688000, 1476304896, 2339064000, 6840038400
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OFFSET
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1,2
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COMMENTS
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All entries greater than 1 are even [Hagis].
14443520 is the first (only?) composite term not divisible by 3. Excluding the factor p=3, all composite terms <= 172972800 have nonincreasing exponents in the factorization (sorted by primes). - D. S. McNeil, Apr 15 2011
Wall shows that 6, 60, and 90 are the only bi-unitary perfect numbers. - Tomohiro Yamada, Apr 15 2017
McNeil's observation about exponents does not hold in general. Indeed, a(41) = 2^8 * 3^5 * 5^2 * 7 * 11 * 13^2 * 17. - Giovanni Resta, Apr 15 2017
We include 1 here, although this is not "multi"-perfect. - R. J. Mathar, Sep 08 2018
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LINKS
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FORMULA
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EXAMPLE
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MATHEMATICA
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bsig[n_] := If[n == 1, 1, Block[{p, e}, Product[{p, e} = pe; (p^(e + 1) - 1)/(p - 1) - If[EvenQ[e], p^(e/2), 0], {pe, FactorInteger[n]}]]]; Select[Range[10^5], Mod[bsig[#], #] == 0 &] (* Giovanni Resta, Apr 15 2017 *)
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PROG
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(PARI) a188999(n) = {my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
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CROSSREFS
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KEYWORD
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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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