OFFSET
1,1
COMMENTS
Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..206
EXAMPLE
3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.
MATHEMATICA
q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 10 2021
STATUS
approved