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A302573 Primitive unitary abundant numbers (definition 1): unitary abundant numbers (A034683) all of whose proper unitary divisors are unitary deficient.

%I A302573 #11 Jun 22 2019 19:02:32
%S A302573 70,840,924,1092,1386,1428,1430,1596,1638,1870,2002,2090,2142,2210,
%T A302573 2394,2470,2530,2970,2990,3190,3230,3410,3510,3770,4030,4070,4510,
%U A302573 4730,5170,5390,5830,13860,15015,16380,17160,18480,19635,20020,21420,21840,21945,22440
%N A302573 Primitive unitary abundant numbers (definition 1): unitary abundant numbers (A034683) all of whose proper unitary divisors are unitary deficient.
%C A302573 The unitary analog of A071395.
%C A302573 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.
%D A302573 J. Sandor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, p. 115.
%H A302573 Amiram Eldar, <a href="/A302573/b302573.txt">Table of n, a(n) for n = 1..10000</a>
%H A302573 V. Siva Rama Prasad and D. Ram Reddy, <a href="http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a58_40.pdf">On primitive unitary abundant numbers</a>, Indian J. Pure Appl. Math., Vol. 21, No. 1 (1990) pp. 40-44.
%e A302573 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.
%t A302573 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]
%Y A302573 Cf. A034448, A034683, A071395, A129487, A302574.
%K A302573 nonn
%O A302573 1,1
%A A302573 _Amiram Eldar_, Apr 10 2018