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A302573
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Primitive unitary abundant numbers (definition 1): unitary abundant numbers (A034683) all of whose proper unitary divisors are unitary deficient.
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5
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70, 840, 924, 1092, 1386, 1428, 1430, 1596, 1638, 1870, 2002, 2090, 2142, 2210, 2394, 2470, 2530, 2970, 2990, 3190, 3230, 3410, 3510, 3770, 4030, 4070, 4510, 4730, 5170, 5390, 5830, 13860, 15015, 16380, 17160, 18480, 19635, 20020, 21420, 21840, 21945, 22440
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OFFSET
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1,1
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COMMENTS
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Prasad & Reddy proved that n is a primitive unitary abundant number if and only if 0 < usigma(n) - 2n < 2n/p^e, where p^e is the largest prime power that divides n.
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REFERENCES
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J. Sandor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, p. 115.
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LINKS
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EXAMPLE
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70 is primitive unitary abundant since it is unitary abundant (usigma(70) = 144 > 2*70), and all of its unitary divisors are unitary deficient. The smaller unitary abundant numbers, 30, 42, 66, are not primitive, since in each 6 is a unitary divisor, and 6 is not unitary deficient.
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MATHEMATICA
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maxPower[n_]:=Max[Power @@@ FactorInteger[n]]; usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; d[n_]:=usigma[n]-2n; punQ[n_] := d[n]>0 && d[n]< 2n/maxPower[n]; Select[Range[1000], punQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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