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

1,1

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.

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.

Example

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.

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]

Crossrefs

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

Sequence in context: A265726 A258375 A306953 * A251933 A061170 A125114

Adjacent sequences: A302570 A302571 A302572 * A302574 A302575 A302576

Keyword

nonn

Author

Amiram Eldar, Apr 10 2018

Status

approved