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

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

Offset: 1

Author

Amiram Eldar, Apr 10 2018

Keywords

• nonn

Comments

The unitary analog of A071395.

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.

Examples

			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.
		

References

• J. Sandor, D. S. Mitrinovic, and B. Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, p. 115.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• V. Siva Rama Prasad and D. Ram Reddy, On primitive unitary abundant numbers, Indian J. Pure Appl. Math., Vol. 21, No. 1 (1990) pp. 40-44.

Crossrefs

Cf. A034448, A034683, A071395, A129487, A302574.

Programs

Mathematica

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]